Chapter 9 of Algebra 1: Concepts and Skills, built for a visual learner. Square roots drawn as square sides, both roots kept on the number line, radicals simplified with factor trees, the anatomy of a parabola, solving by graphing, the quadratic formula read as a centre plus a distance, the discriminant as a root counter, and quadratic inequalities as regions.
Subject: Algebra 1 · 60 slides · symbolic lesson
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Title
Algebra 1 · Chapter 9
Square roots, parabolas, and the one formula that solves every quadratic ever written
Objectives
Straight lines are behind you. A quadratic has a squared term, and its graph is a curve that turns around.
Figure (svg): A parabola with its vertex, axis of symmetry, and two x-intercepts all labelled
Section
Section 9.1
Concept
The square root of a number is the side length of a square with that area. Squaring and rooting are inverse operations.
Figure (svg): A square of area 49 with its side labelled 7, showing that a square root asks for the side length
radical sign — The symbol denoting a square root. Written alone it means the positive root only, which is why solving an equation needs the plus-or-minus sign written in explicitly.
Worked example
Evaluate the exact roots, then estimate one that is not a whole number.
\[ \sqrt{81} \qquad \sqrt{50} \]
For a perfect square, recall the side length
Why: Nine times nine is 81, so the root is 9.
\[ \sqrt{81} = 9 \]
For a non-perfect square, find the two perfect squares it sits between
Why: Fifty lies between 49 and 64, so its root lies between 7 and 8, and much nearer 7.
\[ 7 < \sqrt{50} < 8 \quad \text{so about } 7.07 \]
Figure (svg): A number line marking the perfect squares 49 and 64 with the root of 50 sitting just above 7
Verify: square the estimate
Why: Seven point zero seven squared is about 49.98, which is very close to 50, so the estimate is a good one.
Prediction
Commit before reasoning.
\[ x^2 = 36 \]
Predict first
How many values of x satisfy this equation?
Correct: Two: 6 and negative 6.
Why: Both 6 squared and negative 6 squared give 36, because multiplying two negatives gives a positive. The radical symbol on its own means only the positive root, but the equation asks for every number that squares to 36 — and there are two. Dropping the negative root is the most common quadratic error there is.
Sorting
Sort each number by whether its square root is a whole number.
Sort into buckets
Which of these are perfect squares?
Explain it to yourself
The radical symbol always produces a non-negative number. Say why that is a choice worth making.
Discussion prompt
Why is the radical symbol defined to give only the positive root, when two numbers square to the same value?
Hint: What would go wrong if one symbol stood for two different numbers?
Answer:
Because a symbol has to name exactly one number to be useful. If the radical meant both roots at once, every expression containing one would be ambiguous and could not be computed.
So the symbol takes the positive root, and when a problem genuinely needs both, the plus-or-minus is written in explicitly. That way the ambiguity is visible in the problem rather than hidden in the notation.
Section
Section 9.2
Concept
When a quadratic has no plain x term, you can isolate the squared part and take the root of both sides — remembering both answers.
Figure (svg): A number line showing both minus six and six as solutions of x squared equals thirty six
\[ x^2 = 36 \;\Longrightarrow\; x = \pm 6 \]
Worked example
Solve the equation below.
\[ 3x^2 - 12 = 63 \]
Isolate the squared term
Why: Add 12 to both sides, then divide by 3 — the ordinary two-step moves from Chapter 3.
\[ 3x^2 = 75 \;\Longrightarrow\; x^2 = 25 \]
Take the square root of both sides, writing plus-or-minus
Why: Two numbers square to 25, so the plus-or-minus has to be written or half the answer is lost.
\[ x = \pm 5 \]
Figure (svg): A parabola crossing the horizontal axis at minus five and five, showing both solutions
Verify: substitute both answers into the original
Why: At x equal to 5: 3 times 25 is 75, minus 12 is 63. At x equal to negative 5: the square is still 25, so it gives 63 as well. Both work.
Error analysis
Find what this solution left out.
Annotate
On: \( x^2 = 49 \;\overset{?}{\Longrightarrow}\; x = 7 \)
This is not a small slip. On a graph it means reporting one of the two crossings and missing the other entirely.
Edge cases
Push the method to its edge.
\[ x^2 = -16 \]
Discussion prompt
Why does this equation have no real solutions, and what does its graph look like?
Hint: Can any real number squared come out negative?
Answer:
Squaring any real number gives a result that is zero or positive, so no real number can square to negative 16.
On a graph, the parabola for y equals x squared plus 16 sits entirely above the horizontal axis and never crosses it. No crossings means no real roots — the algebra and the picture agree exactly.
Fill the middle
Fill each blank.
Fill in the blanks
2x^2 + 5 = 55 \;\Longrightarrow\; x^2 = 25 \;\Longrightarrow\; x = \pm 5
Why: Subtracting 5 gives 2x squared equals 50, and dividing by 2 gives x squared equals 25. The root is 5, and because both 5 and negative 5 square to 25, the plus-or-minus is essential. Dividing before subtracting would be a legal but messier route to the same place.
Section
Section 9.3
Concept
A radical is simplified by finding the largest perfect square factor and taking its root outside the sign.
\[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]
Figure (svg): The number 50 factored into 25 times 2, with the 25 escaping the radical as a 5
Worked example
Simplify the expression below.
\[ \sqrt{72} \]
Find the largest perfect square that divides it
Why: Seventy-two is 36 times 2, and 36 is the largest perfect square factor. Choosing a smaller one such as 4 works but leaves more to do.
\[ \sqrt{36 \cdot 2} \]
Split the radical across the product
Why: The root of a product is the product of the roots, so the perfect square can be handled on its own.
\[ \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \]
Figure (svg): A factor tree of 72 branching to 36 and 2, with 36 splitting again into 6 and 6
Verify: square the simplified form
Why: Six root two squared is 36 times 2, which is 72 — the number we started with, so the simplification preserved the value.
Matching
Look for the largest perfect square factor in each.
Match the pairs
Why: Each splits as a perfect square times a small leftover: 9 times 2, 16 times 3, 25 times 3, and 16 times 2. Choosing a smaller square factor still works but leaves the answer unsimplified — for 48, taking 4 gives 2 root 12, which needs simplifying again.
Error analysis
This step looks reasonable and is not legal. Find why.
Annotate
On: \( \sqrt{9 + 16} \;\overset{?}{=}\; \sqrt{9} + \sqrt{16} = 3 + 4 = 7 \)
Whenever a radical sits over a sum, finish the sum before touching the radical.
Discrimination
Sort each expression by whether any work remains.
Sort into buckets
Which of these are in simplest radical form?
Section
Section 9.4
Concept
A quadratic function graphs as a parabola: a symmetric curve with a single turning point called the vertex.
Figure (svg): A parabola with its vertex, axis of symmetry, and two x-intercepts all labelled
\[ y = ax^2 + bx + c \qquad \text{vertex at } x = -\frac{b}{2a} \]
Worked example
Graph the function below, marking its vertex and axis of symmetry.
\[ y = x^2 - 2x - 3 \]
Find the axis of symmetry
Why: Using negative b over 2a: negative of negative 2, over 2 times 1, gives x equal to 1.
\[ x = 1 \]
Find the vertex by substituting that x
Why: One minus two minus three is negative four, so the vertex is at (1, -4).
Build a small table using the symmetry
Why: Points equally far either side of the axis have the same height, so each computation gives two points.
| x | y | mirror point |
|---|---|---|
| 1 | -4 | the vertex |
| 2 | -3 | (0, -3) |
| 3 | 0 | (-1, 0) |
| 4 | 5 | (-2, 5) |
Figure (svg): A parabola with its vertex, axis of symmetry, and two x-intercepts all labelled
Verify: check the symmetry of two mirror points
Why: At x equal to 3 the value is 0, and at x equal to negative 1 it is also 0. Both are 2 units from the axis at x equal to 1, exactly as symmetry requires.
Tweak it
One number controls whether the parabola opens up or down, and how tightly.
Parameter explorer
Drag a. What happens as it passes through zero, and which point never moves?
\[ y = {a}x^2 \]
Prediction
Read the leading coefficient and commit.
\[ y = -2x^2 + 8x - 5 \]
Predict first
Which way does this parabola open, and is its vertex a maximum or a minimum?
Correct: Downward, with the vertex as a maximum.
Why: The leading coefficient is negative 2, and a negative leading coefficient always opens the parabola downward. A downward parabola rises to its turning point and then falls, so the vertex is the highest point — a maximum. The other coefficients shift the curve around but never change which way it opens.
Socratic
One question, no computation.
Discussion prompt
The vertex sits halfway between the two x-intercepts. How does that fact explain the formula for the axis of symmetry?
Hint: What happens if you average the two roots given by the quadratic formula?
Answer:
A parabola is symmetric, so its turning point must be exactly midway between any two points at the same height — including the two x-intercepts.
The quadratic formula gives the two roots as negative b over 2a, plus and minus the same amount. Averaging them cancels the plus-or-minus part entirely and leaves negative b over 2a, which is therefore the midpoint and the axis.
\[ \frac{1}{2}\left(\frac{-b + \sqrt{D}}{2a} + \frac{-b - \sqrt{D}}{2a}\right) = \frac{-b}{2a} \]
Section
Section 9.5
Concept
The solutions of a quadratic equation are the x-intercepts of its graph — the places where the output is zero.
Figure (svg): Three parabolas showing two x-intercepts, one touching point, and no crossing at all
A parabola can cross the axis twice, touch it once, or miss it entirely, and those are the only three possibilities.
Worked example
Solve the equation below by graphing.
\[ x^2 - 2x - 3 = 0 \]
Graph the matching function
Why: The equation asks where the output is zero, so graph y equal to that expression and look for the crossings.
Read the x-intercepts off the picture
Why: The curve crosses at negative 1 and at 3.
\[ x = -1 \quad \text{or} \quad x = 3 \]
Figure (svg): A parabola crossing the horizontal axis at minus one and three, with both crossings circled
Verify: substitute both roots into the original equation
Why: At x equal to 3: 9 minus 6 minus 3 is 0. At x equal to negative 1: 1 plus 2 minus 3 is 0. Both give zero, so both are genuine roots.
Prediction
Look at the picture in your head before computing.
\[ y = x^2 + 4 \]
Predict first
How many times does this parabola cross the horizontal axis?
Correct: Not at all — its lowest point is 4 units above the axis.
Why: The vertex is at (0, 4) and the parabola opens upward, so every output is at least 4 and the curve never reaches zero. The matching equation x squared plus 4 equals zero has no real solutions, which is the algebraic form of the same fact.
Error analysis
A student read the roots of a graph as approximately 1.4 and negative 1.4 and stopped there.
Annotate
On: \( x^2 - 2 = 0 \;\Longrightarrow\; x \approx \pm 1.4 \)
Graph to understand, then use algebra to get the number.
Comparison
Fill the blanks. Each method suits a different shape of problem.
Comparison matrix
| method | best when | limitation |
|---|---|---|
| square roots | there is no plain x term | useless once a bx term appears |
| graphing | you want to see how many roots exist | inexact for irrational roots |
| the quadratic formula | always, for any quadratic at all | more arithmetic than the others |
The formula never fails, which makes it the safe default. The other two are faster when they apply.
Pattern
Four routes, and a decision that takes two seconds.
The first line is the one people skip, and skipping it makes every later line wrong.
Explain it
A classmate asks why a quadratic graph is a mirror image rather than any other curve.
Discussion prompt
Explain in two sentences, using the squared term, why the graph must be symmetric.
Hint: What do 3 squared and negative 3 squared have in common?
Answer:
Squaring destroys the sign, so inputs that are equally far either side of the turning point produce exactly the same output — 2 squared and negative 2 squared are both 4.
That equal-outputs-for-mirrored-inputs property is precisely what symmetry means, so the curve has no choice but to fold onto itself about a vertical line.
Ranking
Estimate each root without a calculator, then order them from smallest to largest.
Put in order
Why: The roots are 2, about 3.16, exactly 5, about 6.32, and exactly 9. Because square roots grow more slowly than the numbers under them, quadrupling the number only doubles the root — which is why 40 sits between 25 and 81 rather than far beyond both.
Section
Section 9.6
Concept
The quadratic formula solves any equation of the form a x squared plus b x plus c equals zero, whatever the numbers.
Figure (svg): The quadratic formula with each part labelled: the discriminant under the root and the plus-or-minus producing two answers
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Worked example
Solve the equation below using the formula.
\[ 2x^2 + 5x - 3 = 0 \]
Identify a, b and c, with their signs
Why: Here a is 2, b is 5 and c is negative 3. Capturing the sign of c is where this usually goes wrong.
Compute the discriminant first
Why: Five squared is 25, and 4 times 2 times negative 3 is negative 24, so subtracting gives 25 plus 24, which is 49.
\[ b^2 - 4ac = 25 + 24 = 49 \]
Substitute into the formula
Why: The root of 49 is 7, and the denominator is 2 times 2, which is 4.
\[ x = \frac{-5 \pm 7}{4} \]
Split the plus-or-minus into two answers
Why: Adding gives 2 over 4, and subtracting gives negative 12 over 4.
\[ x = \tfrac{1}{2} \quad \text{or} \quad x = -3 \]
Figure (svg): The parabola for the equation crossing the axis at minus three and one half
Verify: substitute one root into the original
Why: At x equal to negative 3: 2 times 9 is 18, plus 5 times negative 3 is negative 15, minus 3, giving 0. The root checks out.
Notation
The formula is dense. Decode what each part is doing.
Annotate
On: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Read this way, the formula is not a jumble of symbols — it is a centre plus a distance.
Error analysis
Find the error in this substitution.
Annotate
On: \( 3x^2 - 7x - 6 = 0 \;\overset{?}{\Longrightarrow}\; b^2 - 4ac = 49 - 4(3)(6) = -23 \)
Write a, b and c out with their signs before substituting anything. It costs one line and prevents this entirely.
Faded example
Fill each blank.
Fill in the blanks
x^2 - 6x + 5 = 0 \;\Longrightarrow\; b^2 - 4ac = 16 \;\Longrightarrow\; x = \frac4}}5 \;\Longrightarrow\; x = 1 \text___ ___
Why: With a equal to 1, b equal to negative 6 and c equal to 5, the discriminant is 36 minus 20, which is 16, whose root is 4. The two roots are 6 plus 4 over 2, which is 5, and 6 minus 4 over 2, which is 1. Note that negative b is positive 6 because b itself was negative.
Section
Section 9.7
Concept
The discriminant is the part under the radical. Its sign alone tells you how many real solutions there are, before you solve anything.
\[ b^2 - 4ac \]
Figure (svg): Three parabolas showing two x-intercepts, one touching point, and no crossing at all
Worked example
Without solving, say how many real solutions each equation has.
\[ x^2 + 2x + 5 = 0 \qquad x^2 - 6x + 9 = 0 \]
Compute the first discriminant
Why: Two squared is 4, and 4 times 1 times 5 is 20, so 4 minus 20 is negative 16.
\[ b^2 - 4ac = -16 < 0 \;\Rightarrow\; \text{no real solutions} \]
Compute the second
Why: Thirty-six minus 4 times 1 times 9, which is 36, gives exactly zero.
\[ b^2 - 4ac = 0 \;\Rightarrow\; \text{exactly one solution} \]
Interpret both geometrically
Why: The first parabola never reaches the axis; the second touches it at a single point without crossing.
Figure (svg): Two parabolas, one floating entirely above the axis and one resting on it at a single point
Verify: check the second equation by factoring
Why: x squared minus 6x plus 9 is the square of x minus 3, so the only solution is 3 — a single repeated root, exactly as a zero discriminant predicted.
Sorting
Compute each discriminant and sort. Do not solve.
Sort into buckets
How many real solutions does each equation have?
Prediction
Commit on the count alone.
\[ 3x^2 - 12x + 12 = 0 \]
Predict first
How many real solutions does this have?
Correct: Exactly one — the discriminant is zero.
\[ 3x^2 - 12x + 12 = 3(x - 2)^2 \]
Why: The discriminant is 144 minus 4 times 3 times 12, which is 144 minus 144, giving zero. So the two roots coincide at a single value, x equals 2. Factoring confirms it: the expression is 3 times the square of x minus 2.
Two truths and a lie
Three claims about the discriminant.
Eliminate the wrong options
Which statement is false?
Survives elimination: C
Why: Keep the false statement, which is C. A negative discriminant means no REAL solutions. There are still two solutions in a wider number system you will meet in Algebra 2, and the distinction matters because the equation is not broken — it simply has no answers among the numbers available so far.
Section
Section 9.8
Concept
A quadratic inequality asks where the curve is above or below the axis. The roots divide the number line into regions, and the answer is whole regions.
Figure (svg): A parabola with the region below the curve shaded, showing the solution of a quadratic inequality
\[ x^2 - 4 < 0 \;\Longleftrightarrow\; -2 < x < 2 \]
Worked example
Solve the inequality below.
\[ x^2 - 4 < 0 \]
Find the roots of the matching equation
Why: Setting the expression to zero gives x squared equals 4, so the roots are plus and minus 2.
Sketch the parabola and note which way it opens
Why: The leading coefficient is positive, so it opens upward and dips below the axis between the roots.
Read off where the curve is below the axis
Why: Below the axis means a negative output, which happens strictly between the two roots.
\[ -2 < x < 2 \]
Figure (svg): A parabola with the region below the curve shaded, showing the solution of a quadratic inequality
Verify: test one value from each of the three regions
Why: At x equal to 0 the expression is negative 4, which satisfies it. At x equal to 3 it is 5, which does not. At x equal to negative 3 it is also 5, which does not. Only the middle region works.
Discrimination
The direction of the sign decides which regions survive. Sort each inequality.
Sort into buckets
Where is the solution set for each, given all these parabolas open upward?
Reverse engineer
Work backwards from a pair of solutions.
Fill in the blanks
\text5 2 \text7 5 \;\Longrightarrow\; (x - 2)(x - ___) = 0 \;\Longrightarrow\; x^2 - ___x + 10 = 0
Why: A root of 2 comes from a factor of x minus 2, and a root of 5 from x minus 5. Multiplying out gives x squared minus 7x plus 10, where the 7 is the sum of the roots and the 10 is their product. That relationship between roots and coefficients is a fast way to check any quadratic answer.
Real world
A ball is thrown upward and its height in metres after t seconds is modelled below.
\[ h = -5t^2 + 20t \]
Discussion prompt
When is the ball above 15 metres, and what shape does the answer have?
Hint: Which way does a parabola with a negative leading coefficient open?
Answer:
Setting the height equal to 15 gives negative 5t squared plus 20t equals 15, which rearranges to t squared minus 4t plus 3 equals zero, with roots at t equal to 1 and t equal to 3.
The parabola opens downward, so the ball is above 15 metres between those times: from 1 second to 3 seconds.
Notice how the downward opening reverses the usual rule. Always sketch which way the curve opens before deciding between or outside.
Check
Solve it on paper before you click.
Check your understanding
Solve x^2 - 5x + 6 = 0.
Answer: A
Why: The discriminant is 25 minus 24, which is 1, so the roots are 5 plus or minus 1, all over 2, giving 3 and 2. Checking x equal to 2: 4 minus 10 plus 6 is 0.
Check
Solve it on paper before you click.
Check your understanding
How many real solutions does 4x^2 - 4x + 1 = 0 have?
Answer: A
Why: The discriminant is 16 minus 4 times 4 times 1, which is 16 minus 16, giving zero. A zero discriminant means one repeated root, here x equal to one half. The expression factors as the square of 2x minus 1.
Trap
Solve the equation below with the quadratic formula.
\[ x^2 + 3x = 10 \]
Read the coefficients straight off as they appear
Why: It looks like a is 1, b is 3 and c is 10, since those are the numbers on the page.
\[ b^2 - 4ac = 9 - 40 = -31 \;\Rightarrow\; \text{no real solutions} \]
But there obviously are solutions — x equal to 2 works. The formula was applied to an equation that was not in the right form.
Rearrange to standard form first, then read the coefficients.
Subtract 10 from both sides so the right side is zero
Why: The formula is derived for an equation equal to zero, so it means nothing until the equation is in that form.
\[ x^2 + 3x - 10 = 0 \;\Rightarrow\; c = -10 \]
\[ b^2 - 4ac = 9 + 40 = 49 \;\Rightarrow\; x = 2 \text{ or } x = -5 \]
One rearrangement changes the answer from none to two. Always set it to zero before touching the formula.
Commit first
Answer, then rate your confidence.
\[ y = -x^2 + 6x - 5 \]
Predict first
What is the x coordinate of this parabola's vertex?
Correct: 3
\[ x = \frac{-6}{2(-1)} = 3 \]
Why: Using negative b over 2a: b is 6 and a is negative 1, so it is negative 6 over negative 2, which is 3. The two negatives cancel. Substituting gives a vertex at (3, 4), and since the parabola opens downward that is its maximum height.
Exit ticket
Last commitment of the chapter.
Predict first
Which of these is shakiest right now?
Correct: Whatever you picked is the one to drill first.
Why: The third one causes the most lost marks, and it has a mechanical fix: write a equals, b equals and c equals on three separate lines with their signs before touching the formula. That one habit removes most quadratic-formula errors outright.
Connect it up
One page, drawn by you.
Draw it
Draw one parabola large in the middle and label the vertex, axis of symmetry and roots. Around it, attach: square roots, the plus-or-minus, simplifying radicals, the quadratic formula, the discriminant, and quadratic inequalities. On each arrow, write what that idea tells you about the drawing.
If the discriminant is not connected to how many times the curve crosses the axis, add that arrow — it is the link that makes the whole chapter cohere.
Recap
You can now handle equations whose graphs curve, which is most of the interesting mathematics ahead of you.
| if you remember one thing | it should be |
|---|---|
| about roots | two numbers square to the same value, so write the plus-or-minus |
| about the formula | set the equation to zero first, then write a, b and c with signs |
| about the discriminant | positive gives two, zero gives one, negative gives none |
| about inequalities | sketch which way the parabola opens before choosing a region |
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