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Amy Ferguson Moncure

Algebra 1 · Unit 9

Compare Methods for Solving Quadratics

Choose factoring, square root, completing the square, or the formula from how the equation is set up, and check the solutions.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Look at the setup

Already factored, or easy integer factors: factor. x² = k or (x − h)² = k: square root. a = 1 and you want vertex form too: complete the square. Anything: the formula always works.

Check the answers

Substitute, or compare two methods on the same equation.

Efficiency

Factoring is fastest when the pair is obvious. The formula is fastest when factoring stalls. Completing the square is extra work unless you need the vertex.

Same solutions

A correct method must match the others. If they disagree, an arithmetic slip happened.

Level 1

Easy

Name a good method, then solve.

Example
Which method is a natural first try for (x − 4)² = 9? Solve.

Square root: the square is already isolated.

x − 4 = ±3

x = 7 or x = 1

Answer: square root; x = 1, x = 7

1
Which method for x² + 5x + 6 = 0? Solve.
Show Answer

Factor: 2 and 3.

(x + 2)(x + 3) = 0

x = −2, x = −3

Answer: factor; x = −3, x = −2

2
Which method for x² = 50? Solve.
Show Answer

Square root.

x = ±5√2

Answer: square root; ±5√2

3
Which method always works for 2x² − x − 7 = 0 if you do not see factors?
Show Answer

Quadratic formula.

Answer: formula

4
Solve x² − 16 = 0 two ways: factor, and square root.
Show Answer

(x + 4)(x − 4) = 0

x² = 16, x = ±4

Same solutions.

Answer: x = ±4

5
Why is completing the square clumsy as a first try for (x + 1)(x + 4) = 0?
Show Answer

It is already factored. Set each factor to 0: x = −1, x = −4.

Answer: already factored

Level 2

Medium

Solve two ways, or choose after rewriting.

Example
Solve x² − 6x + 5 = 0 by factoring and by completing the square. Show the answers match.

Factor: (x − 1)(x − 5) = 0, x = 1 or x = 5.

CTS: x² − 6x = −5, (x − 3)² = 4, x − 3 = ±2, x = 5 or x = 1.

Answer: x = 1, x = 5

1
Solve 2x² = 18 by square root and by factoring after moving 18.
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x² = 9, x = ±3

2x² − 18 = 0, 2(x² − 9) = 0, (x + 3)(x − 3) = 0

Answer: x = ±3

2
For x² + 4x − 1 = 0, why is the formula (or CTS) better than factoring over the integers?
Show Answer

No integer pair multiplies to −1 and adds to 4 except none that work for integers.

D = 16 + 4 = 20, not a perfect square of an integer pair in the usual list.

x = −2 ± √5

Answer: −2 ± √5

3
A story width w, area w(w + 3) = 28. Which method, and which root do you keep?
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w² + 3w − 28 = 0, (w + 7)(w − 4) = 0

Factoring is natural. Keep w = 4.

Answer: factor; w = 4

4
Solve (2x + 1)² = 25 by square root. Then expand and factor to check.
Show Answer

2x + 1 = ±5

x = 2 or x = −3

4x² + 4x + 1 − 25 = 0, 4x² + 4x − 24 = 0, x² + x − 6 = 0, (x + 3)(x − 2) = 0.

Answer: x = −3, x = 2

5
When would you complete the square even if the formula is faster for the roots?
Show Answer

When you also want vertex form (h, k) for a graph or a max/min story.

Answer: when you need the vertex

Level 3

Hard

Pick a method from the setup, then justify.

Example
Compare three methods on x² − 4x − 12 = 0: factor, CTS, formula. Which is most efficient here, and what are the solutions?

Factor: (x − 6)(x + 2) = 0, x = 6 or x = −2. Fastest; the pair is obvious.

CTS: (x − 2)² = 16, same roots.

Formula: 4 ± √(16 + 48)2 = 4 ± 82, same roots.

Answer: factor; x = −2, x = 6

1
2x² + 5x − 1 = 0. Why start with the formula rather than factoring over the integers?
Show Answer

D = 25 + 8 = 33, not a perfect square, so the roots are irrational.

x = −5 ± √334

Answer: −5 ± √334

2
A ball h(t) = −16t² + 64t. When does it hit the ground? Name the best method.
Show Answer

h(t) = 0 is −16t(t − 4) = 0 after factoring out −16t.

Factoring (GCF) is best. t = 0 launch, t = 4 landing.

Answer: factor GCF; t = 4 s

3
Solve x² + 10x + 16 = 0 by the method of your choice, then check with another method.
Show Answer

Factor: (x + 2)(x + 8) = 0, x = −2, x = −8

CTS: (x + 5)² = 9, x + 5 = ±3, same roots.

Answer: x = −8, x = −2

4
(x − 3)² = 7. A student expands to x² − 6x + 2 = 0 and uses the formula. Show square root is shorter and matches.
Show Answer

x − 3 = ±√7

x = 3 ± √7

Formula on x² − 6x + 2: 6 ± √(36 − 8)2 = 6 ± √282 = 6 ± 2√72 = 3 ± √7

Answer: x = 3 ± √7

5
Give an equation where completing the square is the method you would teach first, and one where you would not.
Show Answer

First: x² + 8x − 3 = 0, a = 1, half of 8 is 4, and you may want (x + 4)² = 19.

Not first: (x − 1)(x + 5) = 0, already factored.

Answer: CTS when a = 1 and vertex is useful

Super Challenge

Super Challenge

1
For ax² + bx + c = 0, outline a decision: when to factor, when to take square roots, when to complete the square, when to use the formula.
Show Answer

If it is (x − h)² = k or x² = k, square root.

If a pair for c and b is obvious, factor.

If you need vertex form, complete the square.

Otherwise the formula. Always check by substituting.

Answer: see decision list

2
A student gets x = 2 and x = −5 by factoring x² + 3x − 10 = 0, and x = 5 and x = −2 from the formula. Who is wrong, and how do you know?
Show Answer

Check x = 2: 4 + 6 − 10 = 0. Check x = −5: 25 − 15 − 10 = 0. Factoring is right.

The formula slip was likely −b ± √D2a with the wrong sign on b.

−3 ± 72 gives 2 and −5, not 5 and −2.

Answer: formula was wrong; x = −5, 2

3
Show that for x² − 2x − 4 = 0, CTS and the formula are essentially the same arithmetic.
Show Answer

CTS: (x − 1)² = 5, x = 1 ± √5

Formula: 2 ± √(4 + 16)2 = 2 ± √202 = 2 ± 2√52 = 1 ± √5

Completing the square is how the formula is built.

Answer: x = 1 ± √5

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