Quadratic Factored Form Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Factored Form.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The factored form of a quadratic function is f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2), where r1r_1 and r2r_2 are the zeros (roots) of the function and aa is the leading coefficient.

Each factor (x−r)(x - r) equals zero when x=rx = r. So the factored form literally shows you where the parabola crosses the xx-axis—plug in either root and the whole expression becomes zero.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factored form a(x−r1)(x−r2)a(x-r_1)(x-r_2) shows exactly where the parabola hits the x-axis.

Common stuck point: The procedure for quadratic factored form is the easy part; the trap is reading roots with the wrong sign. Asking "Is the quadratic written as a product of linear factors, and do I want where it equals zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the quadratic written as a product of linear factors, and do I want where it equals zero?

Worked Examples

Example 1

easy
What are the zeros of f(x)=(x−1)(x+4)f(x) = (x - 1)(x + 4)?

Answer

x=1 and x=−4x = 1 \text{ and } x = -4

First step

1
Set each factor to zero: x−1=0x - 1 = 0 gives x=1x = 1; x+4=0x + 4 = 0 gives x=−4x = -4.

Full solution

  1. 2
    The zeros are x=1x = 1 and x=−4x = -4.
  2. 3
    The graph crosses the xx-axis at these points.
In factored form a(x−r1)(x−r2)a(x - r_1)(x - r_2), the zeros are read directly as r1r_1 and r2r_2. This is the most convenient form for finding xx-intercepts.

Example 2

medium
Write a quadratic in factored form with zeros at x=3x = 3 and x=−2x = -2 and passing through (0,−12)(0, -12).

Example 3

easy
Find the axis of symmetry of f(x)=(x+2)(x−8)f(x)=(x+2)(x-8).

Example 4

medium
Factor 3x2−123x^2 - 12 completely.

Example 5

medium
Solve (2x−3)(x+5)=0(2x-3)(x+5)=0.

Example 6

hard
Solve 6x2+x−12=06x^2 + x - 12 = 0 by factoring.

Example 7

hard
Find all kk so x2+kx+18=0x^2+kx+18=0 has integer solutions.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the zeros of h(x)=−3(x+2)(x−5)h(x) = -3(x + 2)(x - 5).

Example 2

medium
Find the axis of symmetry of f(x)=(x−1)(x−7)f(x) = (x - 1)(x - 7).

Example 3

easy
Find the roots of (x−2)(x−5)=0(x-2)(x-5) = 0.

Example 4

easy
Find the roots of (x+3)(x−1)=0(x+3)(x-1) = 0.

Example 5

easy
Factor x2−7x+12x^2 - 7x + 12.

Example 6

easy
Factor x2+5x+6x^2 + 5x + 6.

Example 7

easy
Write the factored form with roots 44 and −1-1 (leading coefficient 11).

Example 8

easy
Factor out the GCF: x2−4xx^2 - 4x.

Example 9

easy
Factor the difference of squares x2−16x^2 - 16.

Example 10

easy
What are the roots of 2(x−3)(x+5)=02(x-3)(x+5) = 0?

Example 11

medium
Factor 2x2+7x+32x^2 + 7x + 3.

Example 12

medium
Factor 3x2−10x+83x^2 - 10x + 8.

Example 13

medium
Solve x2−5x=0x^2 - 5x = 0 by factoring.

Example 14

medium
Solve (x−2)(x+4)=7(x-2)(x+4) = 7. (Hint: expand first.)

Example 15

medium
Factor completely: 2x2−182x^2 - 18.

Example 16

medium
A quadratic has roots 33 and 33 and leading coefficient 11. Write factored form.

Example 17

medium
Write the factored form of a parabola with roots −2-2 and 55 passing through (0,−20)(0, -20).

Example 18

medium
Factor x2−x−12x^2 - x - 12.

Example 19

medium
Solve x2+6x=−8x^2 + 6x = -8 by factoring.

Example 20

challenge
Can x2+x+1x^2 + x + 1 be factored over the integers? Justify with the discriminant.

Example 21

challenge
A rectangle's area is x2+7x+10x^2 + 7x + 10. Find its possible dimensions as binomials.

Example 22

challenge
Find kk so that x2−kx+9x^2 - kx + 9 factors as a perfect square (x−r)2(x-r)^2.

Example 23

easy
Find the roots of (x−7)(x+2)=0(x-7)(x+2)=0.

Example 24

easy
Find the roots of x(x−6)=0x(x-6)=0.

Example 25

easy
Write a quadratic in factored form (leading coefficient 11) with zeros −6-6 and 11.

Example 26

medium
Factor x2−3x−18x^2 - 3x - 18.

Example 27

medium
Solve x2−7x+12=0x^2 - 7x + 12 = 0 by factoring.

Example 28

medium
Factor 2x2−5x−32x^2 - 5x - 3.

Example 29

medium
Factor x2+2x−15x^2 + 2x - 15.

Example 30

medium
Solve x2=4x+5x^2=4x+5 by factoring.

Example 31

medium
Find the vertex of f(x)=(x−2)(x−10)f(x)=(x-2)(x-10).

Example 32

hard
Factor 4x2−94x^2 - 9.

Example 33

hard
A garden has area x2+9x+20x^2+9x+20 square feet. Write the dimensions as binomials.

Example 34

hard
Write a quadratic in factored form with roots 12\tfrac{1}{2} and −3-3 and integer coefficients (leading coefficient positive, smallest possible).

Example 35

hard
Factor x2−6xy+9y2x^2 - 6xy + 9y^2.

Example 36

challenge
A parabola has zeros at 11 and 55 and a maximum value of 44. Write it in factored form.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionsfactoring