Practice Quadratic Factored Form in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

easy
Factor x2−7x+12x^2 - 7x + 12.

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

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

Example 10

medium
Factor x2−11x+24x^2 - 11x + 24.

Example 11

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

Example 12

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

Example 13

easy
Factor x2−9x+20x^2 - 9x + 20.

Example 14

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

Example 15

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

Example 16

medium
Write the factored form of the quadratic with roots −3-3 and 44 passing through (0,−24)(0,-24).

Example 17

challenge
For what values of kk does x2+8x+kx^2 + 8x + k factor as (x+r)2(x+r)^2 for some real rr?

Example 18

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

Example 19

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

Example 20

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