Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Solving Quadratic Equations by Factoring

3.8 3.3About 70 minutes

Learning goals

I can use the zero product property to solve a factored equation.
I can rearrange, factor and solve quadratic equations, and verify the roots.
I can explain that the roots of ax² + bx + c = 0 are the zeros of y = ax² + bx + c.
I can solve problems with quadratic equations and reject answers that do not make sense.

Words to know

quadratic equation
An equation that can be written as ax² + bx + c = 0, where a ≠ 0.

root
A solution of an equation. The roots of ax² + bx + c = 0 are the zeros of y = ax² + bx + c.

zero product property
If a product equals zero, then at least one of its factors equals zero.

double root
A root that appears twice, as in (x − 3)² = 0. Its graph touches the x-axis at one point.

verify
Check a solution by substituting it into the original equation and showing the left side equals the right side.

How big is the rink?

A community group is flooding a rectangular outdoor rink. The length must be 3 m more than the width, and they have enough water and boards for an area of 180 m². How wide should it be?

If the width is x metres, the length is x + 3 and

x(x + 3) = 180.

That is a quadratic equation: an equation with an x² term once you expand it. Guessing could take a while. Factoring solves it in a few lines.

The rink problem from start to finish. Each step is explained in this lesson. The root x = −15 is a correct solution of the equation, but it does not make sense as a length.

The zero product property

If A × B = 0, then A = 0 or B = 0 (or both).

This is the key fact. It works only with zero. If A × B = 6, you cannot conclude anything about A or B on their own, because 2 × 3, 1 × 6 and 0.5 × 12 all make 6.

That is why the first step is always to make one side of the equation equal to zero.

Five steps for solving a quadratic equation by factoring. Don't skip the first and last steps.

Roots and zeros are the same numbers

The solutions of a quadratic equation ax² + bx + c = 0 are called its roots. They are exactly the zeros (x-intercepts) of the relation y = ax² + bx + c, because the zeros are the x-values that make y = 0. So you can check your roots on a graph, and you can find zeros by solving an equation.

Worked example

Example 1: Solve and verify

Solve x² + 10x + 16 = 0 by factoring, and verify algebraically.

One side is already 0. Factor: 2 × 8 = 16 and 2 + 8 = 10, so (x + 2)(x + 8) = 0.
Zero product property: x + 2 = 0 or x + 8 = 0.
So x = −2 or x = −8.
Verify x = −2: LS = (−2)² + 10(−2) + 16 = 4 − 20 + 16 = 0 = RS ✓.
Verify x = −8: LS = 64 − 80 + 16 = 0 = RS ✓.

x = −2 or x = −8

Worked example

Example 2: Rearrange first

Solve 2x² − 5x = 3.

Make one side zero: 2x² − 5x − 3 = 0.
Factor by decomposition: ac = −6; −6 and 1. 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3) = (x − 3)(2x + 1).
x − 3 = 0 gives x = 3. 2x + 1 = 0 gives 2x = −1, so x = −1/2.
Verify x = 3: 2(9) − 5(3) = 18 − 15 = 3 ✓. Verify x = −1/2: 2(1/4) − 5(−1/2) = 0.5 + 2.5 = 3 ✓.

x = 3 or x = −0.5

The roots of 2x² − 5x − 3 = 0 are the zeros of y = 2x² − 5x − 3. The graph crosses the x-axis at x = −0.5 and x = 3, which verifies Example 2 graphically.

Worked example

Example 3: Never divide by x

Solve x² = 9x.

Tempting but wrong: dividing both sides by x gives x = 9 and loses a root.
Rearrange: x² − 9x = 0.
Common factor: x(x − 9) = 0.
x = 0 or x = 9. Both check: 0 = 0 ✓ and 81 = 81 ✓.

x = 0 or x = 9

Worked example

Example 4: Difference of squares and perfect squares

Solve (a) 4x² − 49 = 0 and (b) x² − 6x + 9 = 0.

(a) (2x − 7)(2x + 7) = 0, so 2x = 7 or 2x = −7. x = 3.5 or x = −3.5.
(b) (x − 3)² = 0, so x − 3 = 0 and x = 3.
In (b), both factors give the same root. It is called a double root: the graph of y = x² − 6x + 9 touches the x-axis at one point, (3, 0).

(a) x = ±3.5 (b) x = 3 (a double root)

Worked example

Example 5: Back to the rink

Solve x(x + 3) = 180 and answer the question.

Expand and rearrange: x² + 3x − 180 = 0.
Factor: we need product −180 and sum 3. 15 × (−12) = −180 and 15 + (−12) = 3. So (x + 15)(x − 12) = 0.
x = −15 or x = 12.
A width cannot be negative, so reject x = −15.
Width 12 m, length 15 m. Check: 12 × 15 = 180 ✓.

The rink should be 12 m wide and 15 m long.

Watch out

The zero product property only works when one side is 0. (x − 2)(x + 1) = 4 does not mean x − 2 = 4.
Never divide both sides by a variable; factor it out instead.
In word problems, check whether each root makes sense (lengths and times are usually positive), and answer in a sentence with units.

Hands-on activity

Think-of-a-number puzzles

You need: a family member or friend, paper and a pencil.

Ask your partner to secretly pick a whole number from −10 to 10.
They square it, then subtract 5 times the number, and tell you only the result. For example, if they picked 7: 49 − 35 = 14.
Write the equation x² − 5x = 14, rearrange to x² − 5x − 14 = 0, and solve by factoring.
You will get two possible numbers. Ask one yes-or-no question to find out which one they picked.
Swap roles and try different rules, such as "square it and add 3 times the number".

Model answer for the example: (x − 7)(x + 2) = 0, so the number was 7 or −2. Both really work: (−2)² − 5(−2) = 4 + 10 = 14.

Just for fun

The zero product property, explained at home: if the amount of effort you put into your chores times the amount of chores you finished equals zero... at least one of those numbers was zero. Your family can probably guess which one.

Your turn

Solve (x − 4)(x + 9) = 0.

Solve x² − 7x + 12 = 0.

Solve x² + 2x − 35 = 0.

Solve x² − 16x = 0.

Solve 9x² − 25 = 0.

Solve 3x² + 10x + 8 = 0.

Solve x² + 4 = 4x.

Verify that x = −3 is a root of 2x² + 5x − 3 = 0, then find the other root.

More practice: warm-up

Solve x(x − 6) = 0.

Solve (2x − 1)(x + 5) = 0.

Solve x² − 25 = 0.

Solve x² + 5x + 6 = 0.

Solve x² − x − 20 = 0.

More practice: on your own

Solve 2x² + 7x − 15 = 0.

Solve x(x + 3) = 28.

Solve 5x² = 20x.

The product of two consecutive integers is 132. Find the integers.

A rectangular garden is 4 m longer than it is wide and has an area of 96 m². Find its dimensions.

Challenge

Solve (x − 2)² = 2x − 1.

Write a quadratic equation with integer coefficients whose roots are 3 and −2/3.

For which values of k does x² + kx + 16 = 0 have exactly one solution? What is the solution in each case?

Check your understanding

Quiz

Solve (x + 3)(x − 8) = 0.

Solve x² − 2x − 15 = 0.

What is the best first step to solve x² + 6x = 16?

Solve x² = 7x.

What are the roots of 3x² − 12 = 0?

The roots of x² − 5x + 4 = 0 are the same as which feature of the graph of y = x² − 5x + 4?

For parents and teachers

Big idea: Expectation 3.8 asks students to solve quadratic equations with real roots by factoring, the quadratic formula and graphing. This lesson is the factoring method, including the curriculum's sample "solve x² + 10x + 16 = 0 by factoring, and verify algebraically." It also connects roots of an equation to zeros of a graph (3.3).

How to help: Ask your teen to verify every root by substituting it back into the original equation (left side and right side). In word problems, ask "Do both answers make sense here?"

Common difficulties:

Trying to use the zero product property before making one side 0.
Dividing by x and losing the root x = 0.
Sign errors when going from a factor to a root: (x + 8) = 0 gives x = −8.
Keeping impossible answers (negative lengths) in context problems.