Topic 6 of 7

Quadratics: factoring, completing the square, the quadratic formula

Step 11 · Algebra 1 topics

Solve quadratic equations by factoring, by completing the square and with the quadratic formula, write quadratics in vertex form, and use the discriminant.

Solving by factoring

If two numbers multiply to 0, one of them must be 0. So to solve a quadratic, write it as ax² + bx + c = 0, factor, and set each factor to 0.

Example: Factoring

Solve x² − 5x + 6 = 0.

(x − 2)(x − 3) = 0.
x − 2 = 0 or x − 3 = 0.

Answerx = 2 or x = 3

y = x² − 5x + 6 crosses the x-axis at x = 2 and x = 3: the solutions.

Example: Rearrange first

Solve x² = 4x + 21.

Move everything to one side: x² − 4x − 21 = 0.
(x − 7)(x + 3) = 0.

Answerx = 7 or x = −3

Completing the square

Completing the square rewrites x² + bx as a perfect square minus a number, by adding and subtracting (b ÷ 2)².

x^{2} + bx = \left(x + \tfrac{b}{2}\right)^{2} - \left(\tfrac{b}{2}\right)^{2}

Example: Solving by completing the square

Solve x² + 6x − 7 = 0.

Move the number across: x² + 6x = 7.
Half of 6 is 3, and 3² = 9. Add 9 to both sides: x² + 6x + 9 = 16.
(x + 3)² = 16, so x + 3 = 4 or x + 3 = −4.

Answerx = 1 or x = −7

Example: Vertex form

Write y = x² − 8x + 10 in vertex form.

Half of −8 is −4, and (−4)² = 16.
y = (x² − 8x + 16) − 16 + 10.

Answery = (x − 4)² − 6, with vertex (4, −6)

The quadratic formula

Completing the square on ax² + bx + c = 0 in general gives the quadratic formula, which solves every quadratic equation.

x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Example: Using the formula

Solve 2x² + 3x − 2 = 0.

a = 2, b = 3, c = −2.
b² − 4ac = 9 + 16 = 25.
$x = \frac{-3 \pm 5}{4}$

Answer $x = \frac{1}{2}$ or x = −2

Example: Irrational roots

Solve x² − 2x − 1 = 0.

b² − 4ac = 4 + 4 = 8.
$x = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}$

Answer $x = 1 \pm \sqrt{2}$

b² − 4ac	Real solutions
positive	two
zero	one (a repeated root)
negative	none

Try it yourself

Work each one on paper first, then open the answer to check.

Solve x² + 2x − 15 = 0.
Show answer
x = 3 or x = −5
Solve x² = 9x.
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x = 0 or x = 9
Solve x² − 10x + 21 = 0 by completing the square.
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(x − 5)² = 4, so x = 7 or x = 3
Write y = x² + 4x + 1 in vertex form.
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y = (x + 2)² − 3
Solve 3x² − 5x − 2 = 0.
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x = 2 or $x = -\frac{1}{3}$
How many real solutions does x² + 2x + 5 = 0 have?
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None: b² − 4ac = 4 − 20 = −16.