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+cax^2 + bx + c in vertex form a(xβˆ’h)2+ka(x - h)^2 + k by adding and subtracting the value (b2a)2\left(\frac{b}{2a}\right)^2 to create a perfect square trinomial.

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

Showing a random 20 of 50 problems.

Example 1

easy
What value completes the square for x2βˆ’12xx^2 - 12x?

Example 2

hard
Find the vertex of y=βˆ’2x2+8x+1y = -2x^2 + 8x + 1.

Example 3

easy
Write x2βˆ’20x+100x^2 - 20x + 100 as a squared binomial.

Example 4

hard
For what value of cc is x2βˆ’14x+cx^2 - 14x + c a perfect-square trinomial?

Example 5

challenge
Use completing the square to derive the quadratic formula starting from ax2+bx+c=0ax^2 + bx + c = 0 (a≠0a \neq 0).

Example 6

medium
Complete the square: x2βˆ’6x+1x^2 - 6x + 1.

Example 7

medium
Complete the square: x2+5x+1x^2 + 5x + 1 (fractions allowed).

Example 8

medium
Find the vertex of y=x2βˆ’6x+11y = x^2 - 6x + 11.

Example 9

challenge
Derive the vertex of y=ax2+bx+cy = ax^2 + bx + c by completing the square; give the xx-coordinate.

Example 10

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

Example 11

medium
Solve x2+4xβˆ’1=0x^2 + 4x - 1 = 0 by completing the square.

Example 12

easy
Write x2βˆ’14x+49x^2 - 14x + 49 as a squared binomial.

Example 13

medium
Complete the square: x2+12x+20x^2 + 12x + 20.

Example 14

easy
Write x2+4x+4x^2 + 4x + 4 as a squared binomial.

Example 15

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

Example 16

easy
What number completes the square for x2+10x+ _ x^2 + 10x + \,\_\,?

Example 17

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

Example 18

challenge
For what value of cc is x2+10x+cx^2 + 10x + c a perfect-square trinomial?

Example 19

medium
Complete the square: 2x2+8x+52x^2 + 8x + 5.

Example 20

easy
Write x2βˆ’6x+9x^2 - 6x + 9 as a squared binomial.
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