SAT — mathMedium5 min read

Solving Quadratics by Factoring

Solve quadratic equations by factoring for the Digital SAT. Set equations to zero, factor, and apply the zero product property.

Tutor AI Team2026-02-25T01:20:08.788Z

Solving quadratic equations by factoring is one of the most important skills for the Advanced Math section of the Digital SAT. The method relies on the zero product property: if $A \times B = 0$ , then $A = 0$ or $B = 0$ . By expressing a quadratic as a product of factors set equal to zero, you can find the solutions quickly.

Core Concepts

The Zero Product Property

If $ab = 0$ , then $a = 0$ or $b = 0$ (or both).

This is why we set the equation equal to zero before factoring.

Standard Form

A quadratic equation should be in the form:

$ax^2 + bx + c = 0$

before factoring.

Solving by Factoring: Step by Step

Move all terms to one side (set equal to 0).
Factor the quadratic expression.
Set each factor equal to 0.
Solve each equation.

When $a = 1$ : $x^2 + bx + c = 0$

Find two numbers that multiply to $c$ and add to $b$ .

Example: $x^2 + 5x + 6 = 0$

$6 = 2 \times 3$ and $2 + 3 = 5$ ✓

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

$x = -2$ or $x = -3$

When $a \neq 1$ : $ax^2 + bx + c = 0$

Use the AC method: find two numbers that multiply to $ac$ and add to $b$ .

Example: $2x^2 + 7x + 3 = 0$

$ac = 6$ . Numbers: $1$ and $6$ ( $1 + 6 = 7$ ) ✓

$2x^2 + x + 6x + 3 = 0$

$x(2x + 1) + 3(2x + 1) = 0$

$(x + 3)(2x + 1) = 0$

$x = -3$ or $x = -\frac{1}{2}$

GCF First

Always check for a common factor:

$3x^2 - 12x = 0$

$3x(x - 4) = 0$

$x = 0$ or $x = 4$

Don't divide both sides by $x$ — you'll lose the $x = 0$ solution!

Special Forms

Difference of squares: $x^2 - 16 = 0$ → $(x+4)(x-4) = 0$ → $x = \pm 4$

Perfect square: $x^2 - 10x + 25 = 0$ → $(x-5)^2 = 0$ → $x = 5$ (double root)

Strategy Tips

Tip 1: Always Set Equal to Zero First

Don't try to factor $x^2 + 3x = 10$ directly. Rewrite: $x^2 + 3x - 10 = 0$ .

Tip 2: Check for GCF Before Factoring

If every term shares a factor, pull it out first. This simplifies the factoring.

Tip 3: Don't Divide by the Variable

If $x$ is a common factor, factor it out rather than dividing. $x^2 = 5x$ → $x^2 - 5x = 0$ → $x(x-5) = 0$ → $x = 0$ or $x = 5$ .

Tip 4: Check by Substituting

Plug your solutions back into the original equation to verify.

Tip 5: Backsolving Works on SAT

For multiple choice, substitute answer choices. The correct solutions make the equation true.

Worked Example: Example 1

Problem

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

Find two numbers: multiply to 12, add to $-7$ : $-3$ and $-4$ .

$(x - 3)(x - 4) = 0$ → $x = 3$ or $x = 4$

Solution

Worked Example: Example 2

Problem

Solve $x^2 + x - 20 = 0$ .

Numbers: multiply to $-20$ , add to $1$ : $5$ and $-4$ .

$(x + 5)(x - 4) = 0$ → $x = -5$ or $x = 4$

Solution

Worked Example: Example 3

Problem

Solve $6x^2 - x - 2 = 0$ .

$ac = -12$ . Numbers: $3$ and $-4$ ( $3 + (-4) = -1$ ) ✓

$6x^2 + 3x - 4x - 2 = 0$

$3x(2x + 1) - 2(2x + 1) = 0$

$(3x - 2)(2x + 1) = 0$ → $x = \frac{2}{3}$ or $x = -\frac{1}{2}$

Solution

Worked Example: SAT-Style

Problem

If $x^2 = 8x - 15$ , what are the possible values of $x$ ?

$x^2 - 8x + 15 = 0$

$(x - 3)(x - 5) = 0$ → $x = 3$ or $x = 5$

Solution

Worked Example: Example 5

Problem

Solve $5x^2 + 15x = 0$ .

$5x(x + 3) = 0$ → $x = 0$ or $x = -3$

Solution

Practice Problems

Problem 1

Solve $x^2 + 9x + 14 = 0$ .

Problem 2

Solve $x^2 - 3x - 28 = 0$ .

Problem 3

Solve $3x^2 + 5x - 2 = 0$ .

Problem 4

Solve $x^2 - 49 = 0$ .

Problem 5

Solve $4x^2 - 20x = 0$ .

Problem 6

If $x^2 + 2x = 35$ , what is the positive value of $x$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Not setting equal to zero. Factoring $x^2 + 3x = 10$ directly is wrong. Rewrite as $x^2 + 3x - 10 = 0$ .
Dividing by $x$ . This loses the $x = 0$ solution. Factor instead.
Sign errors. If $c > 0$ and $b < 0$ , both factors are negative: $(x - p)(x - q)$ .
Stopping after factoring. Setting factors equal to zero and solving is the final step — don't skip it.
Missing the negative solution. $x^2 = 9$ gives $x = 3$ AND $x = -3$ .

Frequently Asked Questions

What if the quadratic doesn't factor nicely?

Use the quadratic formula or completing the square instead.

How many solutions can a quadratic have?

At most 2. It can have 2 distinct solutions, 1 repeated solution, or no real solutions.

Can I always use factoring?

Only when the quadratic factors over the integers. Otherwise, use the quadratic formula.

What's a double root?

When both factors give the same solution, like $(x-3)^2 = 0$ → $x = 3$ only.

Is factoring faster than the quadratic formula?

Yes, when it works. Factoring takes seconds; the formula takes longer. Try factoring first.

Key Takeaways

✓
Set equal to zero before factoring.
✓
Zero product property: if $AB = 0$ , then $A = 0$ or $B = 0$ .
✓
GCF first — don't skip this step.
✓
Never divide by a variable — factor it out instead.
✓
Check your solutions by substituting back.
✓
If factoring doesn't work, use the quadratic formula as backup.

Tags:

#sat#math#advanced-math#quadratics#factoring

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Core Concepts

The Zero Product Property

Standard Form

Solving by Factoring: Step by Step

When a=1a = 1a=1: x2+bx+c=0x^2 + bx + c = 0x2+bx+c=0

When a≠1a \neq 1a=1: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

GCF First

Special Forms

Strategy Tips

Tip 1: Always Set Equal to Zero First

Tip 2: Check for GCF Before Factoring

Tip 3: Don't Divide by the Variable

Tip 4: Check by Substituting

Tip 5: Backsolving Works on SAT

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?

When $a = 1$ : $x^2 + bx + c = 0$

When $a \neq 1$ : $ax^2 + bx + c = 0$