Using the discriminant to determine the number of solutions of a quadratic equation. Includes locating the discriminant inside the quadratic formula, the three cases for its sign, computing it with correct signs, predicting the number of x-intercepts of a graph, seeing how changing the constant term moves a parabola through all three cases, and answering a yes-or-no question about whether a height is ever reached.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 9 — Quadratic Equations and Functions
Using the Discriminant
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-546 — the lesson these objectives are drawn from
Warm-up
Lesson 9.6 solved equations with the formula. This lesson looks at just one piece of it.
Discussion prompt
Solve x squared minus two x plus one equals nought with the quadratic formula, and say what is unusual about the answer.
Hint: Compute the part under the radical first.
Answer:
\[ x = \dfrac{2 \pm \sqrt{4 - 4}}{2} = \dfrac{2 \pm 0}{2} = 1 \]
The radical evaluated to nought, so adding and subtracting it gave the same result and the two branches collapsed into a single solution. Whatever sits under that radical is doing more work than any other part of the formula, and this lesson is about that expression alone.
Concept
In the quadratic formula, the expression inside the radical is called the discriminant. Its value determines how many real solutions the equation has, before any of them is found.
discriminant — The expression b squared minus four a c, which appears inside the radical of the quadratic formula. Its sign determines whether a quadratic equation has two solutions, one solution, or no real solution.
Positive gives two, nought gives one, negative gives none.
Figure (svg): The discriminant located inside the quadratic formula
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-540
Section
Section 1
Concept
For the equation a x squared plus b x plus c equals nought, a positive discriminant gives two solutions, a discriminant of nought gives one, and a negative discriminant gives no real solution.
These are Lesson 9.1's three cases for square roots.
Figure (svg): The three cases for the sign of the discriminant
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-540 — The Number of Solutions of a Quadratic Equation, and its Study Tip
Picture it
Three signs, three counts.
Figure (svg): How the discriminant's sign reaches the number of solutions
The reasons are all about square roots rather than about quadratics. Once the discriminant is known, the count follows from what Lesson 9.1 already established.
Worked example
This is Example 1 from the textbook.
\[ \text{Find the discriminant of } x^2 - 3x - 4 = 0 \text{ and say how many solutions there are.} \]
Identify the coefficients
Why: Each with its own sign.
\[ a = 1, \; b = -3, \; c = -4 \]
Substitute
Why: Negative three squared, less four times one times negative four.
\[ (-3) ^{2} - 4(1) (-4) \]
Simplify
Why: Nine plus sixteen.
\[ 9 + 16 \]
Read the sign
Why: Twenty-five is positive.
Figure (svg): Two sign traps in computing the discriminant
\[ b^2 - 4ac = 25 > 0 \;\Longrightarrow\; \text{two solutions} \]
Verify: solve it and count
Why: The formula gives three plus or minus five, over two, which is four and negative one — two distinct solutions, as predicted. The discriminant told us the count in one subtraction and the full formula confirmed it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-540
Matching
Only the sign matters.
Match the pairs
Why: Twenty-five and twelve give the same count despite being different sizes, because only the sign is consulted. The size does say something else — whether the roots are rational — but not how many there are.
Worked example
This is Example 2 from the textbook.
\[ \text{Count the solutions of } x^2 - 2x + 1 = 0 \text{ and } 2x^2 - 2x + 3 = 0. \]
Take the first
Why: Four minus four.
\[ b ^{2} - 4 a c = 0 \]
Read the count
Why: The radical vanishes.
Take the second
Why: Four minus twenty-four.
\[ b ^{2} - 4 a c = -20 \]
Read the count
Why: Negative discriminant.
Figure (svg): The three cases for the sign of the discriminant
\[ 0 \;\Longrightarrow\; \text{one}; \qquad -20 \;\Longrightarrow\; \text{none} \]
Verify: check the first by another route
Why: The first equation factors as x minus one, all squared, equal to nought, so its only solution is one — exactly what a discriminant of nought predicted. Lesson 10.7 makes that factoring routine, and the discriminant is what warns you to look for it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 541-541
Trap
\[ b^2 - 4ac = -20 \]
Report that no solutions were found and keep looking
Why: A negative result feels like a failed calculation rather than an answer.
It is the answer. A negative discriminant proves that no real number satisfies the equation, so continuing to search is looking for something that has been shown not to exist.
The discriminant is negative, so the equation has no real solution.
State the conclusion as a finding, with the reason
Why: The computation has settled the question.
Graphically the parabola simply misses the axis, which is a perfectly ordinary thing for a curve to do.
Faded example
Mind the negative c.
Fill in the blanks
x^2 - 3x - 4 = 0: \quad (-3)^2 - 4(1)(-4) = 9 + 16 = 25
Why: The negative c turns the subtraction into an addition and pushes the discriminant well above nought. Reading c as positive four would give nine minus sixteen, and the wrong conclusion that there is no solution.
Sorting
Compute the discriminant for each.
Sort into buckets
Sort each equation by its number of real solutions.
The two with a discriminant of nought are both perfect square trinomials, which is not a coincidence — Lesson 10.7 shows that a discriminant of nought is exactly the condition for that factoring to work.
Socratic
The formula still has a plus-or-minus.
Discussion prompt
Explain why the two branches of the formula give the same answer when the discriminant is nought. Then say where that single solution sits on the graph.
Hint: What is nought plus nought, and nought minus nought?
Answer:
The plus-or-minus applies to the value of the radical, and when that value is nought both branches read as negative b plus nothing, over two a. Adding and subtracting nought are the same operation, so the two answers coincide rather than one of them disappearing.
That single value is negative b over two a, which is exactly the axis of symmetry from Lesson 9.4 — so the one solution sits at the vertex. Graphically the parabola touches the axis at its turning point and turns back, which is the only way a curve can meet a line once without crossing it.
Section
Section 2
Concept
The b squared term is positive whatever the sign of b, because squaring removes it. The four a c term keeps the signs of a and c, so a negative c makes the discriminant larger.
\[ b^2 - 4ac \]
Standard form must come first, or c will be read wrongly.
Figure (svg): Two sign traps in computing the discriminant
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-541 — the coefficient identification steps in Examples 1 and 2
Picture it
Square, then subtract.
Figure (svg): Two sign traps in computing the discriminant
Only one of the two terms is sensitive to a sign change, which means there is only one place to check rather than two.
Worked example
Three negative coefficients at once.
\[ \text{Find the discriminant of } -2x^2 - 5x - 3 = 0. \]
Identify the coefficients
Why: All three are negative.
\[ a = -2, \; b = -5, \; c = -3 \]
Square b
Why: The sign disappears.
\[ 25 \]
Compute four a c
Why: Four times negative two times negative three.
\[ 24 \]
Subtract
Why: Twenty-five less twenty-four.
\[ 1 \]
Figure (svg): Two sign traps in computing the discriminant
\[ b^2 - 4ac = 25 - 24 = 1 > 0 \]
Verify: notice how the two negatives combined
Why: A negative a and a negative c multiply to a positive, so four a c came out positive and was genuinely subtracted. Two negatives in the product cancel, which is why counting the negative signs matters more than tracking them one at a time.
Elimination
For x squared minus three x minus four equals nought.
Eliminate the wrong options
Which line computes the discriminant correctly?
Survives elimination: A
Why: The b squared term is positive and the negative c turns the subtraction into an addition. Options C and D both reach negative seven by different routes and would both wrongly report no solutions.
Worked example
The equation is not yet in standard form.
\[ \text{Find the discriminant of } x^2 + 2x = 3. \]
Write in standard form
Why: Subtract three from each side.
\[ x ^{2} + 2 x - 3 = 0 \]
Identify the coefficients
Why: c is now negative.
\[ a = 1, \; b = 2, \; c = -3 \]
Substitute
Why: Four minus four times one times negative three.
\[ 4 + 12 \]
Read the sign
Why: Sixteen is positive.
Figure (svg): Two sign traps in computing the discriminant
\[ b^2 - 4ac = 4 + 12 = 16 > 0 \]
Verify: see what skipping the rearrangement would give
Why: Reading c as three would give four minus twelve, which is negative eight, and the false conclusion that there is no solution. In fact the equation has the solutions one and negative three, both easy to check, so the rearrangement is not a formality.
Error analysis
The student was finding the discriminant of x squared minus three x minus four equals nought.
Annotate
On: \( \begin{aligned} a = 1, \; b = -3, \; c &= -4 \\ b^2 - 4ac &= -9 - 4(1)(-4) \\ &= -9 + 16 \\ &= 7 \end{aligned} \)
This particular error happens to leave the count unchanged, since seven is positive too, and that is what makes it dangerous. It would give wrong solutions if the work continued into the full formula, and in an equation with a smaller four a c term it would change the count as well.
Faded example
The equation has terms on both sides.
Fill in the blanks
x^2 + 2x = 3 \;\to\; x^2 + 2x - 3 = 0 \;\to\; c = -3, \quad b^2 - 4ac = 4 + 12
Why: Moving the three across changes its sign, and that change flips the discriminant from negative eight to positive sixteen. The count of solutions depends entirely on doing this step first.
Prediction
For any quadratic with a positive leading coefficient.
Predict first
If a is positive and c is negative, how many real solutions are there?
Correct: Always two, since the discriminant must be positive.
\[ a > 0, \; c < 0 \;\Longrightarrow\; -4ac > 0 \;\Longrightarrow\; b^2 - 4ac > 0 \]
Why: With a positive and c negative, four a c is negative, so subtracting it adds a positive amount to b squared, which is already at least nought. The total is therefore strictly positive whatever b happens to be. Graphically this is obvious: a negative c puts the y-intercept below the axis, and an upward parabola that dips below the axis must cross it on both sides.
Socratic
The four a c term is not.
Discussion prompt
Explain why the sign of b never affects the discriminant while the signs of a and c do. Then say what that means for two equations differing only in the sign of b.
Hint: Count how many times each letter appears.
Answer:
The letter b appears only inside a square, and squaring sends a number and its negative to the same value. The letters a and c appear once each in a product, so a sign change in either one flips the sign of that whole term and changes the result.
Two equations differing only in the sign of b therefore have the same discriminant and the same number of solutions. Their solutions are actually mirror images of each other across the y-axis, because negating b reflects the parabola sideways without changing its shape or how far its vertex sits from the axis.
Section
Section 3
Concept
Each solution of a x squared plus b x plus c equals nought is an x-intercept of the graph of the related function. So the discriminant also counts how many times a parabola meets the x-axis.
This is Lesson 9.5's connection, used in reverse.
Figure (svg): The discriminant predicting the number of x-intercepts
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 541-542 — Examples 3 and 4, Find the Number of x-Intercepts
Picture it
The count before the curve.
Figure (svg): The discriminant predicting the number of x-intercepts
Each count came from one subtraction. Drawing the curves afterwards confirms the prediction but adds nothing to it.
Worked example
This is Example 3 from the textbook.
\[ \text{Will the graph of } y = x^2 + 2x - 2 \text{ meet the x-axis in zero, one or two points?} \]
Set y to nought
Why: Intercepts are where the height is nought.
\[ x ^{2} + 2 x - 2 = 0 \]
Identify the coefficients
Why: c is negative.
\[ a = 1, \; b = 2, \; c = -2 \]
Compute the discriminant
Why: Four plus eight.
\[ 12 \]
Read the count
Why: Positive.
Figure (svg): The discriminant predicting the number of x-intercepts
\[ b^2 - 4ac = 12 > 0 \;\Longrightarrow\; \text{two intercepts} \]
Verify: check with the vertex
Why: The axis is at negative one and the vertex height is one minus two minus two, which is negative three. An upward parabola with its vertex below the axis must cross twice, agreeing with the discriminant.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 541-541
Translation
What the graph does at the axis.
Match the pairs
Why: The middle case is the only one where the curve meets the axis without crossing it, and that happens exactly at the vertex. Crossing and touching are worth distinguishing, because only touching gives a repeated solution.
Worked example
This is Example 4 from the textbook.
\[ \text{Do the graphs of } y = x^2 + 2x + 1 \text{ and } y = x^2 + 2x + 3 \text{ meet the x-axis?} \]
Take the first
Why: Four minus four.
\[ b ^{2} - 4 a c = 0 \]
Read the count
Why: The curve touches once.
Take the second
Why: Four minus twelve.
\[ b ^{2} - 4 a c = -8 \]
Read the count
Why: The curve misses entirely.
Figure (svg): The discriminant predicting the number of x-intercepts
\[ 0 \;\Longrightarrow\; \text{one}; \qquad -8 \;\Longrightarrow\; \text{none} \]
Verify: check both with the vertex
Why: Both have their axis at negative one, and the vertex heights are nought and two respectively. On the axis gives one touching point and above it gives none, which is exactly what the discriminants said.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 542-542
Trap
\[ y = x^2 + 2x - 2 \;\Longrightarrow\; a = 1, \; b = 2, \; c = -2, \; \text{but with } y \text{ still present} \]
Read the coefficients off the function as written
Why: The three numbers are visible, so they were used.
Here it happens to work, because y is already isolated. It fails as soon as the function is written another way, such as y minus one equals x squared plus two x, where c is not the visible constant.
\[ 0 = x^2 + 2x - 2 \]
Substitute nought for y and put the result in standard form
Why: Then read the coefficients.
Writing the substitution line costs three seconds and removes the whole class of error.
Faded example
Then read the coefficients.
Fill in the blanks
y = x^2 + 2x + 3 \;\to\; 0 = x^2 + 2x + 3 \;\to\; b^2 - 4ac = 4 - 12 = -8
Why: The substitution turns a function into an equation, and only then do a, b and c mean what the discriminant expects. The negative result says the curve stays entirely above the axis.
Hypothesis
The discriminant only ever gives three counts.
Predict first
What is the largest number of x-intercepts a parabola can have?
Correct: Two, because the formula gives at most two solutions.
A parabola meets any horizontal line, not just the axis, in at most two points.
Why: The formula produces one value for each sign of the plus-or-minus, so at most two distinct numbers can come out however the coefficients are chosen. Geometrically the curve turns exactly once, so each of its two arms can pass a given height at most once. A curve that turned twice could meet a line three times, which is why cubic equations behave differently — and Lesson 10.8 meets exactly that case.
Socratic
The formula would give the answers anyway.
Discussion prompt
Give two reasons for computing the discriminant before solving. Then say when it is the whole answer rather than a preliminary.
Hint: Think about what you learn and what you avoid.
Answer:
First, it tells you what to expect, so an answer with two roots when the discriminant was nought signals an error immediately. Second, it tells you whether the roots will be rational — a perfect-square discriminant means tidy answers and often means the expression factors, which is usually quicker than the formula.
It is the whole answer whenever the question asks how many rather than which — how many times a graph crosses an axis, whether a thrown object ever reaches a height, whether two curves meet. In those cases finding the roots would be extra work whose result is then discarded, and the discriminant answers directly in one line.
Section
Section 4
Concept
Changing the constant term slides a parabola up or down without moving its axis or changing its width. As c increases, an upward parabola passes from two intercepts to one to none.
The one-solution case is the exact boundary.
Figure (svg): The same parabola raised and lowered by changing c
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 542-542 — Example 5, Change the Value of c
Picture it
Only c differs.
Figure (svg): The same parabola raised and lowered by changing c
All three have their vertex at x equal to negative one, because c plays no part in the axis formula. Only the height changes, and the count of crossings changes with it.
Worked example
This is Example 5 from the textbook, using the earlier examples.
\[ \text{Compare } y = x^2 + 2x + c \text{ for } c = -2, \; 1 \text{ and } 3. \]
Take c equal to negative two
Why: Discriminant four plus eight.
\[ 12, \text{ two intercepts} \]
Take c equal to one
Why: Discriminant four minus four.
\[ 0, \text{ one intercept} \]
Take c equal to three
Why: Discriminant four minus twelve.
\[ -8, \text{ none} \]
Describe the effect
Why: Raising c raises the whole curve.
Figure (svg): The same parabola raised and lowered by changing c
\[ c = -2: \; 12; \quad c = 1: \; 0; \quad c = 3: \; -8 \]
Verify: track the vertex height
Why: The vertex is at x equal to negative one in all three, where the value is one minus two plus c, or c minus one. So the vertex heights are negative three, nought and two — below, on, and above the axis, matching the three counts exactly.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 542-542
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Change | Moves the axis? | Changes the discriminant? |
|---|---|---|
| change c | no | yes |
| change b | yes | yes |
| change the sign of b only | yes, it reflects sideways | no, since b is squared |
The last row is the odd one: flipping the sign of b moves the curve but leaves the count of intercepts alone, because the mirror image of a curve crossing twice still crosses twice.
Worked example
Where exactly does the count change?
\[ \text{For which value of } c \text{ does } y = x^2 + 2x + c \text{ touch the axis exactly once?} \]
Set the discriminant to nought
Why: That is the one-solution condition.
\[ 4 - 4 c = 0 \]
Solve for c
Why: Divide by four.
\[ c = 1 \]
Check the vertex
Why: Its height is c minus one.
\[ 0 \]
Interpret
Why: Below one gives two, above gives none.
Figure (svg): The same parabola raised and lowered by changing c
\[ 4 - 4c = 0 \;\Longrightarrow\; c = 1 \]
Verify: test either side of the boundary
Why: At c equal to 0.9 the discriminant is 0.4, positive, so two intercepts; at c equal to 1.1 it is negative 0.4, so none. The boundary really is a single value, and the count changes the instant it is crossed.
Trap
Raising c from negative two to three should shift the graph, so the axis of symmetry moves too.
Assume every change to the equation moves everything
Why: The graph clearly moved, so all its features must have.
The axis is at negative b over two a, and c does not appear in that expression. All three curves turn at x equal to negative one; only their heights differ.
\[ x = \dfrac{-b}{2a} = -1 \quad \text{for every } c \]
Check which letters appear in the formula for the feature
Why: If c is absent, c cannot affect it.
Changing b, by contrast, moves the axis and changes the discriminant at once.
Faded example
One intercept means a discriminant of nought.
Fill in the blanks
y = x^2 + 2x + c: \quad 4 - 4c = 0 \;\Longrightarrow\; c = 1, \quad \text0 c - 1 = ___
Why: The two conditions agree, as they must: a discriminant of nought and a vertex on the axis are the same statement written two ways. That is why either can be used to find the boundary.
Prediction
An upward parabola, c increasing steadily.
Predict first
What happens to the number of x-intercepts?
Correct: Two, then one, then none, and it stays at none.
\[ c = -2: \; 12 \qquad c = 1: \; 0 \qquad c = 3: \; -8 \qquad c = 10: \; -36 \]
Why: Raising c lifts the whole curve, so the vertex rises steadily from below the axis to on it to above it, and once the lowest point is above the axis no further rise can bring it back. The sequence runs the other way for a downward parabola, whose highest point sinks below the axis as c falls. The single-intercept case occupies exactly one value of c, which is why it is so rarely met by accident.
Socratic
Two of the three cases are common.
Discussion prompt
Explain why exactly one solution happens for only a single value of c. Then say why textbook exercises nevertheless contain plenty of them.
Hint: How wide is the boundary?
Answer:
Two solutions occupy a whole range of values of c and none occupies another whole range, while one solution occurs only at the single value separating them. A boundary between two regions has no width, so picking c at random would essentially never land on it.
Exercises contain many because they are constructed rather than sampled: an author who wants a perfect square trinomial, a repeated root, or a tangent line writes the coefficients to make the discriminant nought on purpose. Those are also the cases that matter most in later work, where a repeated root marks the transition between two behaviours, so the attention they get is out of proportion to how often they occur by chance.
Section
Section 5
Concept
When a question asks whether a height is ever reached, or whether two graphs meet, the discriminant answers it directly. Finding the actual solutions would be extra work that is then discarded.
Yes-or-no questions rarely need the roots themselves.
Figure (svg): A stick thrown towards a tree branch, reaching it or not
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-546 — the lesson opener on throwing a stick over a tree branch
Picture it
Two throws, two answers.
Figure (svg): A stick thrown towards a tree branch, reaching it or not
The slower throw produces a negative discriminant, which is the model saying the stick never reaches that height. No time needs to be computed to know that.
Worked example
Campers hanging food from a branch twenty-two feet up.
\[ \text{A stick is thrown up at } 32 \text{ ft/s from } 5 \text{ feet. Does it reach } 22 \text{ feet?} \]
Set up the model
Why: Thrown upwards, so v is positive.
\[ h = -16 t ^{2} + 32 t + 5 \]
Set the height
Why: The branch is at twenty-two feet.
\[ -16 t ^{2} + 32 t + 5 = 22 \]
Write in standard form
Why: Subtract twenty-two.
\[ -16 t ^{2} + 32 t - 17 = 0 \]
Compute the discriminant
Why: 1024 less 1088.
\[ -64 \]
Figure (svg): A stick thrown towards a tree branch, reaching it or not
\[ b^2 - 4ac = -64 < 0 \;\Longrightarrow\; \text{never reaches it} \]
Verify: check with the maximum height
Why: The vertex is at t equal to one second, where the height is negative sixteen plus thirty-two plus five, which is twenty-one feet. The stick peaks a foot below the branch, exactly as the negative discriminant said, and no time of arrival exists to be computed.
Sorting
Some need a count, some need values.
Sort into buckets
Sort each question by whether the discriminant alone answers it.
The pairs differ by a single word — does against when, how many against what. Reading that word before starting decides whether the work takes one line or five.
Worked example
The same branch, a faster throw.
\[ \text{Now thrown up at } 40 \text{ ft/s from } 5 \text{ feet. Does it reach } 22 \text{ feet?} \]
Set up and set the height
Why: Same branch, faster throw.
\[ -16 t ^{2} + 40 t + 5 = 22 \]
Write in standard form
Why: Subtract twenty-two.
\[ -16 t ^{2} + 40 t - 17 = 0 \]
Compute the discriminant
Why: 1600 less 1088.
\[ 512 \]
Read the count
Why: Positive.
Figure (svg): A stick thrown towards a tree branch, reaching it or not
\[ b^2 - 4ac = 512 > 0 \;\Longrightarrow\; \text{reaches it twice} \]
Verify: say what the two times mean
Why: The stick passes twenty-two feet once on the way up and once on the way down, which is why a positive discriminant gives two times rather than one. Its peak is at t equal to 1.25 seconds, at thirty feet, comfortably above the branch.
Trap
\[ -16t^2 + 32t - 17 = 0 \;\Longrightarrow\; t = \dfrac{-32 \pm \sqrt{-64}}{-32} \]
Apply the full quadratic formula to find when the stick reaches the branch
Why: The question was about the stick reaching the branch, so a time was sought.
The radical has no real value, so the work stops there anyway — and it would have stopped one line earlier had the discriminant been computed first. The question asked whether, not when.
\[ b^2 - 4ac = -64 < 0 \;\Longrightarrow\; \text{no, it never reaches it} \]
Read what the question asks before choosing how much to compute
Why: Whether needs a sign; when needs a value.
Answering with the reason rather than just the word no is what makes it a complete answer.
Faded example
The branch is twenty-two feet up.
Fill in the blanks
-16t^2 + 32t + 5 = 22 \;\to\; -16t^2 + 32t - 17 = 0 \;\to\; b^2 - 4ac = -64
Why: The constant becomes five minus twenty-two, which is negative seventeen. A negative discriminant then says the height is never attained, which the peak height of twenty-one feet confirms.
Elimination
For the stick-and-branch model.
Eliminate the wrong options
The discriminant is negative. What follows?
Survives elimination: A
Why: A negative discriminant means no real time satisfies the equation, so the stick is never at that height. Option B is the boundary case, which would happen if the branch were at exactly twenty-one feet — the peak of this throw.
Socratic
Between reaching and not reaching.
Discussion prompt
Describe what a discriminant of nought would mean for the stick and the branch. Then say why that outcome is unlikely to happen in practice.
Hint: Think about the peak of the throw.
Answer:
It would mean the stick reaches exactly the branch's height at exactly one instant — the top of its flight — and then falls back. The peak would graze the branch rather than clear it, which happens only when the maximum height equals the branch height precisely, at twenty-one feet for the slower throw.
In practice that requires the throwing speed to hit one exact value, and any deviation at all sends the stick either over or under. It is the same reason the single-intercept case is a boundary with no width: real throws land in one of the two regions on either side of it, and the boundary itself is a mathematical dividing line rather than an outcome to aim for.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Discriminant | Solutions | The graph |
|---|---|---|
| positive | two | crosses the x-axis twice |
| zero | one | touches the x-axis at the vertex |
| negative | none | never meets the x-axis |
Every row is the same fact in three languages: a sign, a count and a picture. Being able to move between them is most of what this chapter has been building.
Pattern
To determine the number of solutions of any quadratic equation, these five moves cover it.
Step four is where a negative c turns the subtraction into an addition, and getting that right decides the answer in most of the problems that catch people out.
OpenStax Elementary Algebra 2e, §10.3 Solve Quadratic Equations Using the Quadratic Formula §10.3
Check
Mind the negative c.
Check your understanding
What is the discriminant of x squared minus 3x minus 4 = 0?
Answer: A
Why: Negative three squared is nine, and four times one times negative four is negative sixteen, so subtracting it adds sixteen. Nine plus sixteen is twenty-five.
Check
Set y to nought first.
Check your understanding
How many times does the graph of y = x squared + 2x + 3 meet the x-axis?
Answer: A
Why: The discriminant is four minus twelve, which is negative eight, so the equation has no real solutions and the curve never reaches the axis.
Check
The boundary case.
Check your understanding
For which c does y = x squared + 2x + c touch the x-axis exactly once?
Answer: A
Why: Setting the discriminant four minus four c equal to nought gives c equal to one, and then the vertex height, which is c minus one, is nought.
Real world
This is the camping question from the lesson opener. To keep food from bears, campers throw a stick tied to a rope over a high branch, and the stick's height follows the vertical motion model.
Discussion prompt
A stick is thrown upwards at 32 feet per second from a height of 5 feet, towards a branch 22 feet up. Does it get there? Then answer the same question for a throw at 40 feet per second, and say what the discriminant told you in each case.
Hint: Set the model equal to twenty-two.
Answer:
\[ -16t^2 + 32t + 5 = 22 \;\to\; -16t^2 + 32t - 17 = 0 \;\to\; b^2 - 4ac = -64 \]
The negative discriminant says no real time satisfies the equation, so the stick never reaches twenty-two feet. Its peak is at one second, at twenty-one feet — one foot short.
At forty feet per second the discriminant becomes five hundred and twelve, positive, so the stick passes that height twice: once climbing and once falling. Neither question needed an actual time to be computed, and in the first case no time existed to compute — which is precisely the sort of question the discriminant is for.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
For x squared minus 3x minus 4 = 0, how many real solutions are there?
Correct: Two, since 9 + 16 is 25.
\[ (-3)^2 - 4(1)(-4) = 9 + 16 = 25 \]
Why: The constant term is negative four, so four a c is negative sixteen and subtracting it adds, giving nine plus sixteen. Reading c as positive four is what produces the first option and the false conclusion of no solutions, and it is the single commonest error in the lesson. The third option confuses two separate facts: a perfect-square discriminant means the solutions are rational, not that there is one of them — one solution requires a discriminant of exactly nought. The fourth is simply false, as the equation two x squared minus two x plus three equals nought shows, with its discriminant of negative twenty.
Explain it
They computed b squared as negative nine when b was negative three.
Discussion prompt
In no more than four sentences, explain why that term is always positive and where a sign really does matter in the discriminant. Then give them a one-line check.
Hint: How many times does each letter appear?
Answer:
A usable answer: b appears only inside a square, and squaring a negative gives a positive, so b squared is nine rather than negative nine. The place a sign genuinely matters is the four a c term, where a negative c turns the subtraction into an addition and makes the discriminant larger.
The check is to ask whether the constant term is negative. If it is, and the leading coefficient is positive, the discriminant must come out positive and two solutions are guaranteed — so any negative answer is wrong.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Signs are fixed by remembering that b squared ignores its sign while four a c does not. The count is fixed by the three-case rule: positive two, nought one, negative none. Predicting intercepts is fixed by setting y to nought before reading coefficients. The last is fixed by reading whether the question says how many or which. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write the quadratic formula and box the discriminant inside it, then write the three cases beneath as a short table of sign, count and picture. Underneath, compute the discriminant for four equations chosen so that one has a negative b, one a negative c, one both, and one that needs rearranging first — showing the b squared term and the four a c term on separate lines each time. In the middle, draw one set of axes and sketch y equals x squared plus two x plus c for c equal to negative two, one and three, all on the same picture, marking the shared axis of symmetry as a dashed vertical line and writing each discriminant beside its curve. Beneath that, solve for the value of c that makes the discriminant nought and check it against the vertex height. In the lower corner, set up the stick-and-branch problem for two different throwing speeds and write only the discriminants, with a one-line conclusion for each. Finally, in the margin, write which coefficients affect the axis, which affect the count, and which affect both.
Your three curves should share a vertex position and differ only in height. If they are offset sideways as well, the value of c has crept into the axis calculation, where it does not belong.
Recap
Five things, and the first one costs a single subtraction.
| If the question says | Your first move is |
|---|---|
| How many solutions? | Compute the discriminant and read its sign |
| How many x-intercepts? | Set y to nought, then take the discriminant |
| Does it ever reach that height? | Set the model equal to it and take the discriminant |
| Solve the equation | Use the full formula, not just the discriminant |
| For which c is there one solution? | Set the discriminant equal to nought |
Lesson 9.8 returns to graphs and asks a different question: not where a parabola meets a line, but which side of it a region lies on. Quadratic inequalities shade a whole area rather than marking a few points.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.7 Using the Discriminant §9.7, pp. 540-546 — everything on these slides traces back here
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