Practice Completing the Square in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A technique for rewriting ax2+bx+c in vertex form a(x−h)2+k by adding and subtracting the value (b2a)2 to create a perfect square trinomial.

Imagine you have x2+6x and want a perfect square. A perfect square like (x+3)2=x2+6x+9 needs that extra +9. So you add 9 and subtract 9 to keep the expression equal—then group the perfect square part.

Showing a random 20 of 50 problems.

Example 1

easy
What value completes the square for x2−12x?

Example 2

hard
Find the vertex of y=−2x2+8x+1.

Example 3

easy
Write x2−20x+100 as a squared binomial.

Example 4

hard
For what value of c is x2−14x+c a perfect-square trinomial?

Example 5

challenge
Use completing the square to derive the quadratic formula starting from ax2+bx+c=0 (a≠0).

Example 6

medium
Complete the square: x2−6x+1.

Example 7

medium
Complete the square: x2+5x+1 (fractions allowed).

Example 8

medium
Find the vertex of y=x2−6x+11.

Example 9

challenge
Derive the vertex of y=ax2+bx+c by completing the square; give the x-coordinate.

Example 10

medium
Rewrite x2+6x+2 in vertex form by completing the square.

Example 11

medium
Solve x2+4x−1=0 by completing the square.

Example 12

easy
Write x2−14x+49 as a squared binomial.

Example 13

medium
Complete the square: x2+12x+20.

Example 14

easy
Write x2+4x+4 as a squared binomial.

Example 15

easy
Complete the square: rewrite x2+4x in the form (x+h)2+k.

Example 16

easy
What number completes the square for x2+10x+ _ ?

Example 17

medium
After completing the square, x2+12x+50=(x+6)2+c. Find c.

Example 18

challenge
For what value of c is x2+10x+c a perfect-square trinomial?

Example 19

medium
Complete the square: 2x2+8x+5.

Example 20

easy
Write x2−6x+9 as a squared binomial.
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