Solving by Factoring

Factoring is usually the fastest way to solve a quadratic equation — when the quadratic factors. The whole method rests on one fact called the zero product property: if $ab = 0$ , then $a = 0$ or $b = 0$ . A product can only be zero when at least one of its factors is zero.

That property turns one hard equation into two easy ones. Factor the quadratic into two pieces multiplied together, set each piece equal to zero, and solve each little equation on its own. The answers are called the roots, and a quadratic can have up to two of them.

The zero product property

Multiply any two numbers: the only way the result is $0$ is if one of the numbers is $0$ . There is no other pair that does it — $2 \times 3$ , $-5 \times 4$ , $\dfrac{1}{2} \times 100$ all miss zero. So the moment an equation says $(x + 2)(x + 3) = 0$ , you know $x + 2 = 0$ or $x + 3 = 0$ .

This is why the property only helps when one side is exactly zero. $(x + 2)(x + 3) = 6$ tells you almost nothing — plenty of factor pairs multiply to $6$ .

Factor, then split

The routine has four steps. First, get zero alone on one side. Second, factor the quadratic — for $x^{2} + bx + c$ , find two numbers that multiply to $c$ and add to $b$ . Third, set each factor equal to zero. Fourth, solve each small equation.

For $x^{2} + 5x + 6 = 0$ : the numbers $2$ and $3$ multiply to $6$ and add to $5$ , so the equation factors as $(x + 2)(x + 3) = 0$ , and the roots are $x = -2$ and $x = -3$ .

Get zero on one side first

The most common mistake is factoring before the equation equals zero. In $x^{2} + 3x = 10$ , resist the urge to factor the left side as is — the property does not apply with a $10$ on the right. Subtract $10$ from both sides first: $x^{2} + 3x - 10 = 0$ , which factors as $(x + 5)(x - 2) = 0$ .

One more trap: if every term has an $x$ , like $x^{2} - 9x = 0$ , factor the $x$ out instead of dividing it away. Dividing both sides by $x$ silently throws out the solution $x = 0$ .

Worked examples

Example 1: a straightforward factor

Solve $x^{2} + 5x + 6 = 0$ .

Find two numbers that multiply to

6

and add to

5

2 \text{ and } 3

Factor

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

Set each factor equal to zero

x + 2 = 0 \text{ or } x + 3 = 0

Solve each

x = -2 \text{ or } x = -3

Answer: $x = -2$ or $x = -3$

Example 2: move everything to one side first

Solve $x^{2} + 3x = 10$ .

Subtract

10

from both sides to get zero on the right

x^{2} + 3x - 10 = 0

Find two numbers that multiply to

-10

and add to

3

5 \text{ and } -2

Factor

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

Set each factor equal to zero and solve

x = -5 \text{ or } x = 2

Answer: $x = -5$ or $x = 2$

Example 3: factor out the x

Solve $x^{2} - 9x = 0$ .

Both terms share an

x

— factor it out

x(x - 9) = 0

Set each factor equal to zero

x = 0 \text{ or } x - 9 = 0

Solve each

x = 0 \text{ or } x = 9

Answer: $x = 0$ or $x = 9$

Try one yourself

Common questions

What if the quadratic will not factor?

Not every quadratic factors nicely, and that is fine — factoring is just the fastest tool when it works. If you cannot find the factor pair, solve with the square root method, completing the square, or the quadratic formula instead.

Why does one side have to be zero?

Zero is the only number with this superpower: a product equals zero only when a factor does. If the product equals $6$ or any other number, infinitely many factor pairs work and the split into two small equations is not valid.

Can a quadratic have only one root?

Yes. If both factors are the same, like $x^{2} - 6x + 9 = (x - 3)(x - 3) = 0$ , the two small equations give the same answer. $x = 3$ is called a double root.

How do I check my answers?

Substitute each root back into the original equation. For $x = -2$ in $x^{2} + 5x + 6 = 0$ : $4 - 10 + 6 = 0$ . If both roots check, you factored correctly.

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