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## Completing the Square - Sample Math Practice Problems

The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically graded and the difficulty adapts dynamically based on performance. Answers to these sample questions appear at the bottom of the page. This page does not grade your responses.

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### Complexity=1, Mode=simple

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

 1 4x 2 - 9 = 0 Squared form: x = 2 5x 2 - 8 = 0 Squared form: x =

### Complexity=2, Mode=medium

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

 1 x 2 + 16x + 45 = 0 Squared form: x = 2 x 2 + 14x + 33 = 0 Squared form: x =

### Complexity=3, Mode=hard

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

 1 - 2x 2 + 20x - 47 = 0 Squared form: x = 2 - 2x 2 - 16x - 22 = 0 Squared form: x =

### Complexity=1, Mode=simple

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

14x 2 - 9 = 0
Squared form:
x =
Solution
 4x 2 = 9 1. Add 9 (to both sides) x 2 = 9/4 2. Divide by 4 x 2 = 9/4 3. Completing the square is already done since there is no unsquared x term |x| = 3/2 4. Take the square root. Put any radical fractions into standard form. x = 3/2 5. Solve for x in the two cases or x = - 3/2
25x 2 - 8 = 0
Squared form:
x =
Solution
 5x 2 = 8 1. Add 8 (to both sides) x 2 = 8/5 2. Divide by 5 x 2 = 8/5 3. Completing the square is already done since there is no unsquared x term |x| = 2√10/5 4. Take the square root. Put any radical fractions into standard form. x = 2√10/5 5. Solve for x in the two cases or x = - 2√10/5

### Complexity=2, Mode=medium

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

1x 2 + 16x + 45 = 0
Squared form:
x =
Solution
x 2 + 16x = - 45    1. Subtract 45 (from both sides)
x 2 + 16x + 64 = 19    2. Add 64 to complete the square
( x + 8) 2 = 19    3. Rewrite in squared form
|x + 8| = √19    4. Take the square root. Put any radical fractions into standard form.
x + 8 = √19    5. Solve for x in the two cases...
x + 8 = -√19
x =
 - 8 + √19

6. Subtract 8 (first equation)
x =
 - 8 - √19

7. Subtract 8 (second equation)
2x 2 + 14x + 33 = 0
Squared form:
x =
Solution
 x 2 + 14x = - 33 1. Subtract 33 (from both sides) x 2 + 14x + 49 = 16 2. Add 49 to complete the square ( x + 7) 2 = 16 3. Rewrite in squared form |x + 7| = 4 4. Take the square root. Put any radical fractions into standard form. x + 7 = 4 5. Solve for x in the two cases... x + 7 = - 4 x = - 3 6. Subtract 7 (first equation) x = - 11 7. Subtract 7 (second equation)

### Complexity=3, Mode=hard

Given a quadratic zero equation in x, complete the square to solve.

Example:
A squared form: (x + 2)2 = 5

A solution:   x = -3 + √15 / 3,   -3 - √15 / 3
Type   x^2   for x2 and   sqrt(5)   for √5:
A typed squared form:   (x+2)^2 = 5

A typed solution:   -3 + sqrt(15)/3,   -3 - sqrt(15)/3

1- 2x 2 + 20x - 47 = 0
Squared form:
x =
Solution
 - 2x 2 + 20x = 47 1. Add 47 (to both sides) x 2 - 10x = - 47/2 2. Divide by -2 x 2 - 10x + 25 = 3/2 3. Add 25 to complete the square ( x - 5) 2 = 3/2 4. Rewrite in squared form |x - 5| = √6/2 5. Take the square root. Put any radical fractions into standard form. x - 5 = √6/2 6. Solve for x in the two cases... x - 5 = -√6/2 x = 5 + √6/2 7. Add 5 (first equation) x = 5 - √6/2 8. Add 5 (second equation)
2- 2x 2 - 16x - 22 = 0
Squared form:
x =
Solution
- 2x 2 - 16x = 22    1. Add 22 (to both sides)
x 2 + 8x = - 11    2. Divide by -2
x 2 + 8x + 16 = 5    3. Add 16 to complete the square
( x + 4) 2 = 5    4. Rewrite in squared form
|x + 4| = √5    5. Take the square root. Put any radical fractions into standard form.
x + 4 = √5    6. Solve for x in the two cases...
x + 4 = -√5
x =
 - 4 + √5

7. Subtract 4 (first equation)
x =
 - 4 - √5

8. Subtract 4 (second equation)