Evaluating and approximating square roots. Includes the definition of a square root, the positive and negative roots and the plus-or-minus notation, how many square roots a number has, perfect squares against irrational roots, evaluating radical expressions with the radical bar as a grouping symbol, and handling plus-or-minus expressions on a calculator.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 9 — Quadratic Equations and Functions
Square Roots
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 9.1 Square Roots §9.1, pp. 499-504 — the lesson these objectives are drawn from
Warm-up
You have squared numbers since Chapter 1. This lesson runs that operation backwards.
Discussion prompt
Name every number whose square is 25. Then name every number whose square is negative four.
Hint: Try a negative one as well as a positive one.
Answer:
\[ 5^2 = 25 \quad \text{and} \quad (-5)^2 = 25 \]
Both five and negative five square to twenty-five, so a positive number has two square roots. Nothing squares to a negative, because a negative times a negative is positive and a positive times a positive is too — which is why negative numbers have no real square roots at all.
Concept
If b squared equals a, then b is a square root of a. Squaring and taking a square root are inverse operations, so a square root answers the question of what was squared.
square root — A number b is a square root of a number a when b squared equals a. Three squared is nine, so three is a square root of nine; negative three squared is also nine, so negative three is a square root of nine as well.
All positive real numbers have two square roots.
Figure (svg): Two numbers whose square is nine
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 499-499
Section
Section 1
Concept
Square roots are written with a radical symbol, and the number beneath it is the radicand. A bare radical means the positive square root; a minus sign in front means the negative one; a plus-or-minus sign means both.
\[ \sqrt{9} = 3, \quad -\sqrt{9} = -3, \quad \pm\sqrt{9} = \pm 3 \]
The positive root is also called the principal square root.
Figure (svg): Three ways of writing a square root of nine
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 499-499 — Example 1, Read Square Root Symbols, and the Reading Algebra note on the plus-or-minus symbol
Picture it
Symbol, radicand, expression.
Figure (svg): The parts of a radical expression named
The vocabulary matters more than it looks. Every rule in Lessons 9.3 and 12.2 is stated in terms of radicands, so the words are worth learning now.
Worked example
This is Example 1 from the textbook.
\[ \text{Write } \sqrt{9} = 3, \; -\sqrt{9} = -3 \text{ and } \pm\sqrt{9} = \pm 3 \text{ in words.} \]
Read the bare radical
Why: No sign in front means the positive root.
Read the negative
Why: A minus sign selects the other root.
Read the plus-or-minus
Why: This one names both at once.
\[ \text{both square roots of } 9\text{ are } 3\text{ and } -3 \]
Note the pattern
Why: The symbol never changes; only the sign in front does.
Figure (svg): Three ways of writing a square root of nine
\[ \sqrt{9} = 3, \quad -\sqrt{9} = -3, \quad \pm\sqrt{9} = \pm 3 \]
Verify: square each answer
Why: Three squared is nine and negative three squared is nine, so both values really are square roots of nine. Squaring the answer is the check that works for every square root problem in this chapter.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 499-499
Matching
The sign in front does the work.
Match the pairs
Why: All four share the idea that squaring the answer returns the radicand. Zero is the exception that has only one root, since it is neither positive nor negative.
Worked example
This is Example 2 from the textbook.
\[ \text{Evaluate } \sqrt{64}, \; -\sqrt{64}, \; \pm\sqrt{64} \text{ and } \sqrt{0}. \]
Take the positive root
Why: Eight squared is sixty-four.
\[ \sqrt{64} = 8 \]
Take the negative root
Why: The same radicand, the other sign.
\[ -\sqrt{64} = -8 \]
Take both
Why: The plus-or-minus form.
\[ \pm\sqrt{64} = \pm 8 \]
Take the root of zero
Why: Nought squared is nought.
\[ \sqrt{0} = 0 \]
Figure (svg): A square whose area is sixty-four and whose side is eight
\[ 8, \; -8, \; \pm 8, \; 0 \]
Verify: check the odd one out
Why: Zero is the only number with exactly one square root, because nought is its own negative. Every other perfect square in this example produced a pair, and that asymmetry is worth noticing now.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500
Trap
\[ \sqrt{25} = \pm 5 \]
Give both numbers, since both square to twenty-five
Why: The number really does have two square roots.
It does, but the symbol asks for one of them. A bare radical is defined to mean the positive square root, so writing plus-or-minus answers a question that was not asked.
\[ \sqrt{25} = 5 \quad \text{and} \quad \pm\sqrt{25} = \pm 5 \]
Write the plus-or-minus sign when both roots are wanted
Why: The notation distinguishes the two questions.
This matters from Lesson 9.2 onwards, where solving an equation needs both roots and the formula needs only one.
Faded example
Name which root the symbol asks for.
Fill in the blanks
A radical with no sign in front means the positive square root, and a minus sign in front means the negative square root.
Why: The radicand is unchanged in both cases; only the sign written in front of the symbol differs. Confusing that sign with the sign of the radicand is a separate and much larger error.
Elimination
About the number 49.
Eliminate the wrong options
Which statement is correct?
Survives elimination: A
Why: The number has two square roots and the notation offers three ways of referring to them. Keeping the number's roots separate from the symbol's meaning is the point of this whole section.
Socratic
The number has two roots.
Discussion prompt
Explain why the radical symbol is defined to mean only the positive root. Then say what would go wrong if it meant both.
Hint: Ask whether an expression should name one number.
Answer:
An expression is supposed to name a single number, so that it can be substituted, compared and computed with unambiguously. If the radical named two numbers at once, then a formula containing it would not have a single value, and simple statements like a certain quantity being greater than another would stop making sense.
Defining the symbol as the positive root keeps every radical expression single-valued, and the plus-or-minus sign is then available whenever both roots really are wanted. That is exactly how the quadratic formula uses it in Lesson 9.6 — the formula writes the plus-or-minus explicitly, precisely because the radical alone would not supply it.
Section
Section 2
Concept
Positive real numbers have two square roots. Zero has exactly one, which is zero. Negative numbers have no real square roots, because the square of every real number is positive or zero.
The square root of negative sixty-four is undefined over the reals.
Figure (svg): Two numbers whose square is nine
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500 — the Number of Square Roots rule and the Reading Algebra note on undefined roots
Picture it
Positive, zero, negative.
Figure (svg): Two numbers whose square is nine
The reason is the same in all three cases: squaring a real number can never give a negative result, so nothing is available to be a square root of one.
Worked example
Three radicands, three different answers.
\[ \text{How many real square roots have } 100, \; 0 \text{ and } -100? \]
Take a hundred
Why: Ten and negative ten both square to it.
Take nought
Why: Only nought squares to nought.
Take negative a hundred
Why: No real number squares to a negative.
State the notation
Why: The last one has no value in the real numbers.
Figure (svg): Two numbers whose square is nine
\[ \pm\sqrt{100} = \pm 10, \quad \sqrt{0} = 0, \quad \sqrt{-100} \text{ undefined} \]
Verify: test the negative case directly
Why: Ten squared is a hundred and negative ten squared is also a hundred, so neither gives negative a hundred. Every real number, positive or negative, squares to something positive or nought, so nothing is left over to square to a negative.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500
Sorting
Look only at the radicand.
Sort into buckets
Sort each number by how many real square roots it has.
Whether the roots are whole numbers never entered into it. Two and a quarter both have two roots even though only one of those roots is tidy.
Worked example
Being clear about what the minus sign is attached to.
\[ \text{Evaluate } -\sqrt{16} \text{ and } \sqrt{-16}. \]
Read the first
Why: The minus sits outside the radical.
\[ -\sqrt{16} \]
Evaluate it
Why: Take the root, then negate.
\[ -4 \]
Read the second
Why: The minus sits inside, in the radicand.
\[ \sqrt{-16} \]
Evaluate it
Why: Nothing real squares to negative sixteen.
Figure (svg): The solution to Worked example a minus sign in two places shown as a ladder of expressions, one row per algebraic move
\[ -\sqrt{16} = -4, \qquad \sqrt{-16} \text{ is undefined} \]
Verify: square the first answer
Why: Negative four squared is positive sixteen, so negative four is genuinely a square root of sixteen. The second expression has no answer to square, which is what undefined means here.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500
Trap
\[ \sqrt{-16} = -\sqrt{16} = -4 \]
Pull the minus sign out of the radicand
Why: It worked when factoring, so it looks safe here.
Squaring the result checks it: negative four squared is positive sixteen, not negative sixteen. The minus never came out; it was simply discarded.
\[ \sqrt{-16} \text{ is undefined over the reals} \]
Check whether the radicand is negative before doing anything else
Why: A negative radicand ends the problem.
Later mathematics extends the numbers so that such roots exist, but nothing in this course does.
Prediction
Every real number squared.
Predict first
What is the smallest value that the square of a real number can take?
Correct: Zero, reached only when the number itself is zero.
\[ (-3)^2 = 9, \; (-1)^2 = 1, \; 0^2 = 0, \; 1^2 = 1, \; 3^2 = 9 \]
Why: Squaring a positive gives a positive and squaring a negative also gives a positive, so nought is the floor and only nought reaches it. That single fact explains all three cases in this section: positives are hit twice, nought once, and negatives never.
Elimination
Watch where the minus sign sits.
Eliminate the wrong options
Which of these has no real value?
Survives elimination: A
Why: Only a negative radicand causes trouble. A minus sign in front of the radical is harmless, and telling the two placements apart is the whole skill here.
Socratic
Every other number has two roots or none.
Discussion prompt
Explain why zero has exactly one square root rather than two. Then say what happens to the two roots of a small positive number as it shrinks towards nought.
Hint: Ask what the negative of nought is.
Answer:
Every positive number has a positive root and a negative one, and those are different numbers. Nought's two roots would be nought and its negative, but the negative of nought is nought itself, so the pair collapses into a single value.
As a positive number shrinks, its two roots move towards each other from both sides — the roots of 0.01 are a tenth and negative a tenth, and the roots of 0.0001 are a hundredth and negative a hundredth. They meet exactly when the number reaches nought, which is why nought is the boundary between having two roots and having none.
Section
Section 3
Concept
The square of an integer is a perfect square, and a square root of a perfect square is an integer. If a positive integer is not a perfect square, its square root is irrational and can only be approximated.
perfect square — The square of an integer. The first few are 1, 4, 9, 16, 25, 36, 49 and 64, and a square root of any of them is a whole number.
An irrational number is one that is not a quotient of integers.
Figure (svg): Perfect squares marked on a number line with irrational roots between them
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500 — the definition of a perfect square and Example 3 on exact and approximate values
Picture it
Roots of perfect squares are exact.
Figure (svg): Perfect squares marked on a number line with irrational roots between them
The root of two sits between one and two and never resolves into a fraction, however far it is computed. Lesson 12.9 proves this rather than merely asserting it.
Worked example
This is Example 3 from the textbook.
\[ \text{Evaluate } \sqrt{49} \text{ and } \sqrt{3}, \text{ exactly if possible and otherwise to the nearest hundredth.} \]
Check the first radicand
Why: Forty-nine is seven squared.
Give the exact value
Why: No rounding is needed.
\[ \sqrt{49} = 7 \]
Check the second radicand
Why: Three is not the square of any integer.
Approximate
Why: To the nearest hundredth.
\[ \sqrt{3} \approx 1.73 \]
Figure (svg): Perfect squares marked on a number line with irrational roots between them
\[ \sqrt{49} = 7, \qquad \sqrt{3} \approx 1.73 \]
Verify: square the approximation
Why: 1.73 squared is 2.9929, which is close to three but not equal to it. That gap is the price of rounding, and it is why the exact answer is written with the radical symbol whenever exactness matters.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 500-500
Sorting
Is the radicand a perfect square?
Sort into buckets
Sort each square root by whether it has an exact integer value.
Knowing the perfect squares up to about two hundred makes this instant. Everything else in the list has to be left as a radical or rounded.
Worked example
Estimating before reaching for a calculator.
\[ \text{Between which two consecutive integers does } \sqrt{23} \text{ lie?} \]
Find the perfect square below
Why: Sixteen is four squared.
\[ 16 < 23 \]
Find the perfect square above
Why: Twenty-five is five squared.
\[ 23 < 25 \]
Take roots across the inequality
Why: Larger radicands give larger roots.
\[ 4 < \sqrt{23} < 5 \]
Refine
Why: Twenty-three is close to twenty-five.
\[ \sqrt{23} \approx 4.8 \]
Figure (svg): The solution to Worked example trap a root between two integers shown as a ladder of expressions, one row per algebraic move
\[ 4 < \sqrt{23} < 5, \quad \sqrt{23} \approx 4.80 \]
Verify: square the estimate
Why: 4.8 squared is 23.04, just above twenty-three, so the true root is a shade under 4.8. Squaring an estimate tells you not only whether it is close but on which side it falls.
Error analysis
The student was asked for the exact value of the square root of thirty-six plus the square root of two.
Annotate
On: \( \begin{aligned} \sqrt{36} + \sqrt{2} &= 6 + 1.41 \\ &= 7.41 \\ \text{so the exact value is } &7.41 \end{aligned} \)
Exact and approximate are different requests, and a question usually says which it wants. Writing a radical in a final answer is the normal way of being exact about an irrational number, and Lesson 9.3 is entirely about tidying such answers.
Faded example
Use the nearest perfect squares.
Fill in the blanks
49 < 55 < 64 \;\Longrightarrow\; 7 < \sqrt8 < ___
Why: Taking roots preserves the order, so the root of fifty-five sits between seven and eight. Since fifty-five is nearer sixty-four, the root is nearer eight — about 7.4.
Translation
Exact where possible.
Match the pairs
Why: Two of these are exact and two are rounded, and the word about is doing real work in the latter pair. Writing an equals sign where about belongs is a small dishonesty that compounds through a long calculation.
Socratic
A decimal looks more like an answer.
Discussion prompt
Give two reasons for leaving an irrational square root in radical form. Then say when rounding is the right thing to do.
Hint: Ask what happens when the value is used again.
Answer:
First, the radical is exact and a decimal is not, so an answer written with a radical is correct while a rounded one is merely close. Second, rounding errors grow when a value is used in further calculations, so a rounded root fed into another step produces an answer that is wronger than the one you started with.
Rounding is right at the very end, when a number is being reported for a practical purpose — a length to cut, a cost to pay, a measurement to record — and the question says how precisely. The rule of thumb is to work exactly and round once, at the last moment, rather than at every step.
Section
Section 4
Concept
An expression written with a radical symbol is a radical expression. The radical symbol is a grouping symbol, so everything beneath it must be evaluated to a single number before the root is taken.
\[ \sqrt{b^2 - 4ac} \]
The bar does the same job as a pair of brackets.
Figure (svg): The radical symbol acting as a grouping symbol
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501 — Example 4, Evaluate a Radical Expression, and the note that the radical symbol is a grouping symbol
Picture it
The bar covers the whole expression.
Figure (svg): The radical symbol acting as a grouping symbol
This expression is the discriminant, and it returns in Lessons 9.6 and 9.7 as the heart of the quadratic formula. Evaluating it correctly now pays off twice.
Worked example
This is Example 4 from the textbook.
\[ \text{Evaluate } \sqrt{b^2 - 4ac} \text{ when } a = 1, \; b = -2 \text{ and } c = -3. \]
Substitute the values
Why: Watch the signs on b and c.
\[ \sqrt{(-2)^2 - 4(1)(-3)} \]
Square and multiply
Why: Negative two squared is four; the product is negative twelve.
\[ \sqrt{4 - (-12)} \]
Add
Why: Subtracting a negative adds.
\[ \sqrt{16} \]
Take the positive root
Why: Sixteen is a perfect square.
\[ 4 \]
Figure (svg): The radical symbol acting as a grouping symbol
\[ \sqrt{(-2)^2 - 4(1)(-3)} = \sqrt{16} = 4 \]
Verify: check the sign handling
Why: Four minus negative twelve is four plus twelve, which is sixteen. Reading it as four minus twelve would give negative eight and an undefined root, so the double negative is where the whole example lives.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501
Elimination
Simplifying a radical of a sum.
Eliminate the wrong options
Which move is correct?
Survives elimination: A
Why: The only legal first move is to add underneath the bar. The result is five, and every other option produces a different number, which is the quickest way to see that they cannot all be valid.
Worked example
Guided Practice 13 and 14, with different signs to watch.
\[ \text{Evaluate } \sqrt{b^2 - 4ac} \text{ for } a = 2, b = 3, c = -5 \text{ and for } a = 1, b = -8, c = -20. \]
Take the first set
Why: Nine minus four times two times negative five.
\[ \sqrt{9 + 40} \]
Evaluate it
Why: Forty-nine is a perfect square.
\[ 7 \]
Take the second set
Why: Sixty-four minus four times one times negative twenty.
\[ \sqrt{64 + 80} \]
Evaluate it
Why: A hundred and forty-four is a perfect square.
\[ 12 \]
Figure (svg): The solution to Worked example the same expression, new values shown as a ladder of expressions, one row per algebraic move
\[ \sqrt{49} = 7, \qquad \sqrt{144} = 12 \]
Verify: notice what the negative c does
Why: In both cases c was negative, so subtracting four a c added a positive amount and the radicand grew. A negative c always makes this expression larger, which Lesson 9.7 turns into a statement about how many solutions an equation has.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501
Trap
\[ \sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7 \]
Split the radical across the addition
Why: Multiplication splits this way, so addition looks as though it should.
Nine plus sixteen is twenty-five and the root of twenty-five is five, not seven. The bar groups the sum, and a root cannot be distributed over addition.
\[ \sqrt{9 + 16} = \sqrt{25} = 5 \]
Add underneath the bar, then take one root
Why: The grouping is not optional.
Roots do split over multiplication, which is exactly the rule Lesson 9.3 is built on — but never over addition.
Faded example
Inside the bar first.
Fill in the blanks
a = 1, b = -2, c = -3: \quad \sqrt12 = \sqrt4}} = \sqrt___ = ___
Why: Both the squaring of a negative and the subtraction of a negative product push the value upwards. Getting either sign wrong here changes sixteen into something else, and often into a negative radicand.
Hypothesis
Test it before deciding.
Predict first
Is the square root of a sum equal to the sum of the square roots?
Correct: No, and a single example is enough to show it.
\[ \sqrt{9 + 16} = 5 \quad \text{but} \quad \sqrt{9} + \sqrt{16} = 7 \]
Why: The root of nine plus sixteen is the root of twenty-five, which is five, while the sum of the roots is three plus four, which is seven. One counterexample disproves a general claim, and this one is small enough to carry in your head as a permanent reminder.
Socratic
It is drawn rather than written.
Discussion prompt
Explain what would be ambiguous if the radical bar did not group. Then say which other symbols in algebra group without using brackets.
Hint: Ask where the radicand would end.
Answer:
Without the grouping, an expression like the root of b squared minus four a c would leave it unclear whether the root applies to b squared alone or to the whole difference. The bar's width settles it visually: whatever it covers is inside, and everything else is outside.
A fraction bar does the same job, grouping its whole numerator and its whole denominator without brackets, and so does the horizontal bar in a repeating decimal. All three are cases where the layout carries meaning that would otherwise need punctuation, which is why they must be copied carefully rather than retyped in a line.
Section
Section 5
Concept
When a plus-or-minus sign precedes a radical, the expression represents two different numbers. Compute it twice, once with each sign, and report both results.
A calculator needs the numerator completed before dividing.
Figure (svg): A plus-or-minus expression producing two separate values
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501 — Example 5, Use a Calculator to Evaluate an Expression
Picture it
The same expression, split.
Figure (svg): A plus-or-minus expression producing two separate values
This is the shape of every answer the quadratic formula gives. Getting comfortable with it now makes Lesson 9.6 a matter of substitution rather than of notation.
Worked example
This is Example 5 from the textbook.
\[ \text{Evaluate } \dfrac{1 \pm \sqrt{12}}{4}, \text{ rounding to the nearest hundredth.} \]
Evaluate the radical
Why: The root of twelve, to several places.
\[ \sqrt{12} \approx 3.4641 \]
Take the plus branch
Why: One plus that, all over four.
\[ 4.4641 \div 4 \]
Round it
Why: To the nearest hundredth.
\[ 1.12 \]
Take the minus branch
Why: One minus that, all over four.
\[ -0.62 \]
Figure (svg): A plus-or-minus expression producing two separate values
\[ \dfrac{1 + \sqrt{12}}{4} \approx 1.12, \qquad \dfrac{1 - \sqrt{12}}{4} \approx -0.62 \]
Verify: add the two results
Why: They sum to about 0.5, which is one over four — the radical cancels when the branches are added, since one gains it and the other loses it. That sum is a fast check that both branches were computed from the same expression.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501
Faded example
Same numerator, two signs.
Fill in the blanks
\dfrac1.12}-0.62 \approx ___ \quad \text___ \quad ___
Why: The minus branch is negative because the root of twelve is larger than one, so the numerator comes out negative. Assuming both branches are positive is a common slip when the radical is bigger than the term in front of it.
Worked example
Where the brackets have to go on a calculator.
\[ \text{Why does typing } 1 + \sqrt{12} \div 4 \text{ give the wrong value?} \]
Read what the calculator does
Why: Division before addition, by the order of operations.
\[ 1 + (\sqrt{12} \div 4) \]
Evaluate that
Why: About one plus 0.866.
\[ 1.866 \]
Compare with the intended value
Why: The correct branch is 1.12.
Fix it
Why: Bracket the whole numerator.
\[ (1 + \sqrt{12}) \div 4 \]
Figure (svg): The solution to Worked example a keystroke order that goes wrong shown as a ladder of expressions, one row per algebraic move
\[ (1 + \sqrt{12}) \div 4 \approx 1.12 \]
Verify: check the bracketed version
Why: One plus 3.4641 is 4.4641, and dividing by four gives 1.116, which rounds to 1.12 as expected. The fraction bar grouped the numerator on paper and the brackets have to do that job when the expression is typed in a line.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 501-501
Trap
\[ \dfrac{1 \pm \sqrt{12}}{4} \approx 1.12 \]
Compute the plus branch and stop
Why: The expression produced a number, so it looks finished.
The plus-or-minus sign asks for both. Half the answer has been left out, and in Lesson 9.6 that missing half is a genuine second solution of an equation.
\[ \dfrac{1 \pm \sqrt{12}}{4} \approx 1.12 \text{ and } -0.62 \]
Compute both branches and report both
Why: The sign is an instruction to do the work twice.
Writing the two values side by side from the start is a habit worth forming now.
Prediction
Add the plus branch and the minus branch.
Predict first
What happens to the radical when the two values are added?
Correct: It cancels, leaving twice the other term over the denominator.
\[ 1.12 + (-0.62) = 0.50 = \tfrac{1}{2} \]
Why: One branch adds the radical and the other subtracts it, so the sum keeps only the term in front — here one plus one, over four, which is a half. That cancellation is a genuinely useful check, and in Lesson 9.6 it becomes the fact that the two solutions of a quadratic average to a value determined by b and a alone.
Elimination
Computing the plus branch of the example.
Eliminate the wrong options
Which order gives 1.12?
Survives elimination: A
Why: The fraction bar groups the whole numerator, so brackets are needed when the expression is typed on one line. Two of the wrong options come from grouping too little and one from grouping too much.
Socratic
Two answers could just be written out.
Discussion prompt
Say why the plus-or-minus sign is worth having rather than writing both expressions in full. Then name a place later in this chapter where it is essential.
Hint: Ask how much of the two expressions is shared.
Answer:
The two branches differ in exactly one character, so writing them out separately duplicates everything else and doubles the chances of a copying error. The compact form keeps the shared structure visible and makes it obvious that the two values are related rather than independent.
The quadratic formula in Lesson 9.6 is stated with a plus-or-minus sign for exactly this reason: one formula covers both solutions of every quadratic equation. Lesson 9.2 needs it sooner still, where solving by taking square roots produces two answers from a single step.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Written | Means | Value for 25 |
|---|---|---|
| a bare radical | the positive square root | 5 |
| a minus in front | the negative square root | -5 |
| plus or minus in front | both square roots | 5 and -5 |
The radicand is the same in all three rows and only the sign in front changes. Keeping that separate from the sign of the radicand itself is the distinction this lesson turns on.
Pattern
Whether the radicand is a number or an expression, these five moves cover it.
Step one is where most errors happen, because a mishandled double negative there turns a perfectly good problem into an undefined one.
OpenStax Elementary Algebra 2e, §9.1 Simplify and Use Square Roots §9.1
Check
Read the sign in front.
Check your understanding
What is the value of the negative square root of 81?
Answer: A
Why: The minus sign in front selects the negative root, and negative nine squared is eighty-one. The radicand itself is positive, so the expression is perfectly well defined.
Check
Simplify underneath first.
Check your understanding
Evaluate the square root of b squared minus 4ac when a = 1, b = -2 and c = -3.
Answer: A
Why: Negative two squared is four, and subtracting four times one times negative three adds twelve, giving sixteen. The positive square root of sixteen is four.
Check
How many real roots?
Check your understanding
Which number has exactly one real square root?
Answer: A
Why: Zero is its own negative, so its two roots coincide into the single value nought. Every other non-negative number has a distinct pair.
Real world
This is the chessboard question from the lesson opener. A chessboard is a large square made up of 64 small squares.
Discussion prompt
How many squares are along each side? Then decide whether a board of 100 small squares and a board of 50 small squares can each be built as a square, and say what square roots have to do with the answer.
Hint: The side length is a square root of the total.
Answer:
\[ \sqrt{64} = 8 \quad \text{and} \quad \sqrt{100} = 10 \]
A square board with a whole number of squares along each side needs its total to be a perfect square, so sixty-four gives eight a side and a hundred gives ten.
Fifty is not a perfect square: its root is about 7.07, which lies between seven and eight, so no whole number of squares along a side gives fifty in total. That is a genuine use of the perfect-square idea rather than a decoration on it, and the negative root is discarded here because a side length cannot be negative.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Which of these expressions has no real value?
Correct: The square root of negative 49.
\[ -\sqrt{49} = -7 \qquad \sqrt{-49} \text{ undefined} \]
Why: Only a negative radicand is undefined over the real numbers, because every real number squares to something positive or nought and so nothing squares to negative forty-nine. The first option is simply negative seven: the minus sign sits outside the radical, where it is harmless, and negative seven squared is positive forty-nine. The third option names both seven and negative seven, and the fourth is nought, the one number with exactly one square root. The whole question is about where the minus sign is standing, which is why it is worth slowing down over.
Explain it
They wrote that the square root of nine plus sixteen is seven.
Discussion prompt
In no more than four sentences, explain what went wrong and give them a rule they can apply next time. Then give them the check that would have caught it.
Hint: Ask what the bar covers.
Answer:
A usable answer: the bar of the radical acts like a pair of brackets, so nine and sixteen have to be added before the root is taken. That gives the root of twenty-five, which is five, and a square root cannot be split across an addition.
The check is to square the answer and see whether the radicand comes back. Seven squared is forty-nine rather than twenty-five, which shows immediately that seven was not a square root of the right number.
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: The sign question is fixed by asking whether the minus is under the bar or in front of it. Exactness is fixed by knowing the perfect squares up to about two hundred. Multi-term radicands are fixed by simplifying completely underneath before touching the root. The plus-or-minus form is fixed by writing both branches down before evaluating either. 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 draw a radical expression large and label the radical symbol, the radicand and the whole expression. Beneath it write the same radicand three times, with no sign, a minus and a plus-or-minus in front, and give the value of each. In the middle, draw a number line from nought to ten and mark the roots of every perfect square on it, then mark the roots of two, three, five and seven between them in a second colour, writing beside the line one sentence saying why the second group can never be written exactly as fractions. Underneath, work the discriminant expression for three different sign patterns of a, b and c, showing the substitution and the simplification underneath the bar as separate lines. In the lower corner write out both branches of a plus-or-minus expression and add them together to see the radical cancel. Finally, in the margin, write the three cases for how many real square roots a number can have.
Your number line should show the perfect-square roots getting further apart as you move right. If they look evenly spaced, check the roots of thirty-six and forty-nine against the roots of one and four.
Recap
Five things, and the first is a definition worth stating precisely.
| If the question says | Your first move is |
|---|---|
| Evaluate a bare radical | Take the positive root only |
| A minus sits in the radicand | It is undefined over the reals |
| Give the exact value | Leave the radical if it is irrational |
| Evaluate a radical expression | Simplify everything under the bar first |
| A plus-or-minus sign appears | Compute two values and report both |
Lesson 9.2 puts these roots to work. Taking a square root of both sides of an equation solves a whole family of quadratics in one move — and it is there that the plus-or-minus sign stops being notation and starts being a second solution.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 9 Quadratic Equations and Functions — Lesson 9.1 Square Roots §9.1, pp. 499-504 — everything on these slides traces back here
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