Algebra 1 · Algebra 2 · Grades 9, 10

Solving Quadratic Equations by Factoring

Quick answer

To solve a quadratic by factoring, first write it so one side equals zero, then factor the other side, then set each factor equal to zero and solve. For x squared minus 5x plus 6 = 0, factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3. This works because a product can only be zero if one of its factors is zero.

What you'll learn

  • Solve a quadratic equation by factoring
  • Explain why the equation must be set equal to zero before factoring
  • Recognise the difference of squares and common trinomial patterns

What this method does

A quadratic equation contains an x2x^2 term. Factoring rewrites it as two brackets multiplied together, which turns one hard equation into two much simpler ones:

x2−5x+6=0⟶(x−2)(x−3)=0⟶x=2 or x=3x^2 - 5x + 6 = 0 \quad\longrightarrow\quad (x - 2)(x - 3) = 0 \quad\longrightarrow\quad x = 2 \text{ or } x = 3

When it works it is the fastest method available. It does not always work, and knowing when to stop trying is part of the skill.

Why setting it equal to zero matters

This step gets skipped constantly, and skipping it makes everything after it meaningless. The method rests on one fact:

Zero product property

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

Read it as: the only way to multiply and land on zero is for something you multiplied to have been zero. Nothing else does it. 7×0.00017 \times 0.0001 is tiny but not zero.

That is a remarkably strong claim, and zero is the only number it holds for. Compare:

ab=12ab = 12

Here aa could be 1,2,3,4,6,12,0.5,−3,…1, 2, 3, 4, 6, 12, 0.5, -3, \ldots — knowing the product is 1212 tells you almost nothing about aa on its own.

So if you factor x2−5x+6=12x^2 - 5x + 6 = 12 into (x−2)(x−3)=12(x-2)(x-3) = 12 and then write x−2=12x - 2 = 12, you are using a rule that does not exist. Move everything to one side first, then factor.

How to solve by factoring

  1. Rearrange so one side is 00.
  2. Factor the other side completely.
  3. Set each factor equal to zero.
  4. Solve each small equation.
  5. Check both answers in the original equation.

The patterns worth recognising

PatternFactors asExample
Common factorax2+bx=x(ax+b)ax^2 + bx = x(ax + b)x2+3x=x(x+3)x^2 + 3x = x(x + 3)
Difference of squaresa2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3)
Trinomial, a=1a = 1x2+bx+c=(x+p)(x+q)x^2 + bx + c = (x + p)(x + q) where pq=cpq = c and p+q=bp + q = bx2−5x+6=(x−2)(x−3)x^2 - 5x + 6 = (x-2)(x-3)

For the third pattern you are hunting a pair of numbers that multiply to cc and add to bb. Start from the multiplication, because a number has only so many factor pairs, while endless pairs add to bb.

Worked examples

Common mistakes

Practice problems

  1. Solve x2−7x+12=0x^2 - 7x + 12 = 0.

    Hint

    Find two numbers that multiply to 1212 and add to −7-7.

    Answer

    x=3x = 3 or x=4x = 4

    Full solution

    The pair is −3-3 and −4-4, so (x−3)(x−4)=0(x - 3)(x - 4) = 0.

    Setting each factor to zero gives x=3x = 3 or x=4x = 4.

    Check: 9−21+12=09 - 21 + 12 = 0 ✓ and 16−28+12=016 - 28 + 12 = 0 ✓

  2. Solve x2+5x+6=0x^2 + 5x + 6 = 0.

    Answer

    x=−2x = -2 or x=−3x = -3

    Full solution

    Two numbers multiplying to 66 and adding to 55: 22 and 33. So (x+2)(x+3)=0(x + 2)(x + 3) = 0.

    x=−2x = -2 or x=−3x = -3. Check: 4−10+6=04 - 10 + 6 = 0 ✓

  3. Solve x2−16=0x^2 - 16 = 0.

    Hint

    No middle term, and both parts are perfect squares.

    Answer

    x=4x = 4 or x=−4x = -4

    Full solution

    Difference of squares: (x−4)(x+4)=0(x - 4)(x + 4) = 0, so x=4x = 4 or x=−4x = -4.

  4. Solve x2=9xx^2 = 9x.

    Hint

    Move everything to one side. Do not divide by xx.

    Answer

    x=0x = 0 or x=9x = 9

    Full solution

    x2−9x=0x^2 - 9x = 0, so x(x−9)=0x(x - 9) = 0.

    x=0x = 0 or x=9x = 9. Dividing by xx at the start would have lost x=0x = 0.

  5. Solve x2+2x−15=0x^2 + 2x - 15 = 0.

    Answer

    x=3x = 3 or x=−5x = -5

    Full solution

    Two numbers multiplying to −15-15 and adding to 22: 55 and −3-3. So (x+5)(x−3)=0(x + 5)(x - 3) = 0.

    x=−5x = -5 or x=3x = 3. Check: 9+6−15=09 + 6 - 15 = 0 ✓

  6. Solve x2−10x+25=0x^2 - 10x + 25 = 0.

    Answer

    x=5x = 5 only

    Full solution

    The pair is −5-5 and −5-5, so (x−5)2=0(x - 5)^2 = 0 and x=5x = 5 is a repeated root.

    Check: 25−50+25=025 - 50 + 25 = 0 ✓

  7. Solve x2+3x=10x^2 + 3x = 10.

    Answer

    x=2x = 2 or x=−5x = -5

    Full solution

    Subtract 1010: x2+3x−10=0x^2 + 3x - 10 = 0. The pair is 55 and −2-2, so (x+5)(x−2)=0(x + 5)(x - 2) = 0.

    x=−5x = -5 or x=2x = 2. Check: 4+6=104 + 6 = 10 ✓

  8. Will x2+2x+5=0x^2 + 2x + 5 = 0 factor with whole numbers? Explain how you can tell before trying.

    Hint

    Compute the discriminant b2−4acb^2 - 4ac first.

    Answer

    No — the discriminant is −16-16, so there are no real solutions.

    Full solution

    b2−4ac=22−4(1)(5)=4−20=−16b^2 - 4ac = 2^2 - 4(1)(5) = 4 - 20 = -16.

    A negative discriminant means the equation has no real solutions, so no pair of real factors exists and searching for one would be wasted effort.

Frequently asked questions

Why does the equation have to equal zero before I factor?

Because the whole method rests on the fact that a product is zero only when one of its factors is zero. If the product equals 12, there are endless pairs that multiply to 12, so knowing the factors tells you nothing. Zero is the one number that pins the factors down.

What if the quadratic will not factor with whole numbers?

Then use the quadratic formula or complete the square. Factoring only finds solutions that happen to be rational. If you cannot spot a factor pair within a minute, switch methods rather than hunting.

Can a quadratic have only one solution?

Yes. If the two factors are identical, as in (x - 3)(x - 3) = 0, both give x = 3. That single value is called a repeated or double root, and its graph touches the x-axis instead of crossing it.

What to learn next

Key terms in this lesson

Quadratic equation
A quadratic equation is one that can be written as ax squared plus bx plus c = 0, with a not equal to zero. The squared term is what makes it quadratic rather than linear.
Zero product property
The zero product property says that if a product equals zero, at least one of its factors must be zero. It is the reason factoring solves quadratic equations.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.HSA.SSE.B.3aSeeing Structure in ExpressionsFactor a quadratic expression to reveal the zeros of the function it defines.
  • CCSS.MATH.CONTENT.HSA.REI.B.4bReasoning with Equations and InequalitiesSolve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
For teachersTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.