How to Solve a Quadratic Equation by Factoring

Problem

x^2 - 5x + 6 = 0

Answer

x = 2,\ x = 3

Factoring is the fastest way to solve a quadratic when the expression splits into two neat brackets. The method uses one key fact: a product is zero only when a factor is zero.

Step-by-step solution

Find two numbers that multiply to 6 and add to -5.
$-2 \times -3 = 6,\quad -2 + -3 = -5$
Why: These two numbers give the middle term and the constant term.
Write the factored form.
$(x - 2)(x - 3) = 0$
Why: Expanding the brackets returns the original expression.
Set each bracket equal to zero.
$x - 2 = 0 \quad \text{or} \quad x - 3 = 0$
Why: A product is zero only when at least one factor is zero.
Solve each small equation.
$x = 2 \quad \text{or} \quad x = 3$
Why: Each value makes one bracket zero.

Common mistakes

Using 2 and 3 instead of -2 and -3. The sum must be -5.
Setting the whole product to 6 instead of 0. The rule needs 0.
Stopping at one solution. A quadratic can have two.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$x^2 - 7x + 12 = 0$

Show answer

$x = 3,\ x = 4$
$x^2 + 5x + 6 = 0$

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$x = -2,\ x = -3$
$x^2 - x - 6 = 0$

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$x = 3,\ x = -2$
$x^2 - 9 = 0$

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$x = 3,\ x = -3$
$x^2 + 7x + 10 = 0$

Show answer

$x = -5,\ x = -2$