Solve quadratic equations by factoring

Ontario Grade 11 Mathematics

Quadratic Functions

Use factors and the zero product property to find the values that make a quadratic equation true.

A quadratic equation can have two solutions, one solution, or no real solutions. In this lesson, you will solve quadratic equations by factoring. The central idea is to rewrite an equation as a product equal to zero. Then you can find the values that make at least one factor zero. You will use familiar skills: expanding brackets, finding common factors, and solving simple equations.

What you will learn

Recognize a quadratic equation and write it with zero on one side.
Factor a quadratic expression into simpler expressions.
Use the zero product property to find and check solutions.

1. Prerequisite bridge: expressions, factors, and equations

An expression is a mathematical phrase without an equals sign. For example,

x^2+5x+6

is an expression. An equation has an equals sign, such as

x^2+5x+6=0

.

A factor is a quantity multiplied by another quantity. In

3(x+2)

, the factors are

3

and

(x+2)

. Factoring reverses expanding brackets: it rewrites an expression as a product of factors.

A quadratic expression has a variable raised to the power of two as its highest power. A quadratic equation sets a quadratic expression equal to another expression. In this lesson, the equations will be rearranged so that one side is zero.

x^2+5x+6=(x+2)(x+3)

Expanding multiplies factors; factoring rewrites an expression as a product.
A quadratic equation includes a squared variable as its highest power.

2. The main idea: make a product equal zero

Consider the equation

(x-4)(x+1)=0

. A product is zero when at least one of its factors is zero. So either

x-4=0

or

x+1=0

. These give

x=4

or

x=-1

.

This rule is called the zero product property. It says that if two or more factors multiply to zero, at least one factor must be zero. It lets us turn a factored equation into simpler equations.

Do not use this property on an expression that is not a product equal to zero. For example,

x(x+2)=5

is not ready for the property because the product is not equal to zero. First, put all terms on one side so that the other side is zero.

(x-4)(x+1)=0\Rightarrow x-4=0 or x+1=0

Set one side of the equation equal to zero before using the zero product property.
Set each factor equal to zero, then solve each resulting equation.

3. Factoring and reading the solutions

A common type of quadratic is

x^2+bx+c

, where

b

and

c

are numbers. To factor it, look for two numbers whose product is

c

and whose sum is

b

. Those numbers become the constants in the two brackets.

For instance, in

x^2+5x+6

, the numbers

2

and

3

multiply to

6

and add to

5

. So the expression factors as

(x+2)(x+3)

. The factors reveal the solutions when the expression is set equal to zero.

Some quadratics have a number in front of

x^2

. In those cases, the factors may need more than one number in each bracket. You can test a factorization by expanding it. The expanded expression must match the original expression, term by term.

After finding possible solutions, check each one in the original equation. Substituting a solution means replacing the variable with that value. A correct solution makes both sides equal.

x^2+bx+c=(x+m)(x+n), m+n=b, mn=c

For a monic trinomial, choose constants that multiply to the last term and add to the middle coefficient.
Expand a proposed factorization to check that it matches the original expression.
Substitute solutions into the original equation to check them.

4. A reliable solving routine

First, arrange the equation so that one side is zero. Combine like terms if needed. Like terms have the same variable part, such as

3x

and

-5x

.

Next, factor the quadratic expression. Look for a common factor first if every term shares one. Then check the factors by expanding.

Set each factor equal to zero and solve. Finally, substitute the results into the original equation. This last check can catch a sign error or an incorrect factorization.

ab=0\Rightarrow a=0 or b=0

Rearrange, factor, set each factor to zero, solve, and check.
A solution is a value of the variable that makes the original equation true.

Checking the factorization

Part	Product from the brackets	Result
First terms	$(3x)(x)$	$3x^2$
Outer and inner terms	$(3x)(-2)+(1)(x)$	$-6x+x=-5x$
Constant terms	$(1)(-2)$	$-2$

Worked example

Solve a quadratic with a coefficient in front of the squared term

Solve

3x^2-5x-2=0

by factoring.

Check the equation form
The equation already has zero on one side, so it is ready to factor.
$3x^2-5x-2=0$
Factor the expression
We need factors that expand to $3x^2-5x-2$ . The brackets $(3x+1)$ and $(x-2)$ work: their first terms produce $3x^2$ , their constant terms produce $-2$ , and the middle terms combine to $-5x$ .
$3x^2-5x-2=(3x+1)(x-2)$
Use the zero product property
Since the product equals zero, at least one bracket must equal zero. Solve one equation for each bracket. 3x+1=0 or x-2=0
Solve each equation
Subtracting or adding within each simple equation gives the two possible values of $x$ . x=- $\frac{1}{3}$ or x=2
Check the values
Substitution into the original equation gives zero for each value. For $x=2$ , the terms total $12-10-2=0$ . For $x=-\frac{1}{3}$ , they total $\frac{1}{3}+\frac{5}{3}-2=0$ .
$3(2)^2-5(2)-2=0$

Answer: The solutions are

x=2

and

x=-\frac{1}{3}

.

Check: Both values make the original left side equal zero.

Common mistakes and how to avoid them

Setting each term of a sum equal to zero.

Correction: The zero product property applies to factors multiplied together, not to separate terms being added. Factor the expression first.

Using the zero product property before making one side zero.

Correction: Rearrange the equation so the factored product equals zero before setting its factors equal to zero.

Changing a sign while solving a factor equation.

Correction: Solve each factor equation carefully, then substitute each result into the original equation.

Giving only one solution.

Correction: Check every factor. Each factor can give a solution, though the resulting values may sometimes be the same.

Lesson summary

A quadratic equation can be solved by rewriting its quadratic expression as a product of factors.
Arrange the equation with zero on one side before applying the zero product property.
Set each factor equal to zero, solve the simpler equations, and check the answers in the original equation.

Check your understanding

Question 1

Which pair of factors correctly rewrites

x^2+7x+12

?

$(x+3)(x+4)$
$(x-3)(x-4)$
$(x+2)(x+6)$
correctIndex

Show answer and explanation

(x+3)(x+4)

The constants

3

and

4

multiply to

12

and add to

7

. Expanding the first choice gives

x^2+7x+12

.

Question 2

What values solve

(x-5)(x+2)=0

?

$x=5$ or $x=-2$
$x=-5$ or $x=2$
$x=5$ or $x=2$
correctIndex

Show answer and explanation

x=5

or

x=-2

Set each factor equal to zero:

x-5=0

gives

x=5

, and

x+2=0

gives

x=-2

.

Question 3

What should you do first to solve

x^2+3x=10

by factoring?

Rewrite it as $x^2+3x-10=0$ .
Set $x^2=0$ and $3x=10$ .
Divide both sides by $x$ .
correctIndex

Show answer and explanation

Rewrite it as

x^2+3x-10=0

.

Move all terms to one side to make an equation equal to zero. Then factor and use the zero product property.

Key terms

Quadratic equation: An equation whose highest power of the variable is two.
Factor: A quantity multiplied by another quantity to form a product.
Factoring: Rewriting an expression as a product of factors.
Zero product property: If a product equals zero, at least one of its factors must equal zero.
Solution: A value of the variable that makes an equation true.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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