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Completing the Square for the Vertex - 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

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

1.  
y = 2x 2 + 19
y =
vertex =
2.  
y = 5x 2 + 8
y =
vertex =

Complexity=2, Mode=medium

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

1.  
y = x 2 + - 8x + 30
y =
vertex =
2.  
y = x 2 + - 4x + 15
y =
vertex =

Complexity=3, Mode=hard

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

1.  
y = 5x 2 + - 60x + 194
y =
vertex =
2.  
y = 4x 2 + - 72x + 327
y =
vertex =

Answers

Complexity=1, Mode=simple

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

#ProblemCorrect AnswerYour Answer
1y = 2x 2 + 19
y =
vertex =
Solution
The equation is already simplified. In y = a(x - h)2 + k form here,
h = 0 so (x - h) is just x, and k = 19.
#ProblemCorrect AnswerYour Answer
2y = 5x 2 + 8
y =
vertex =
Solution
The equation is already simplified. In y = a(x - h)2 + k form here,
h = 0 so (x - h) is just x, and k = 8.

Complexity=2, Mode=medium

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

#ProblemCorrect AnswerYour Answer
1y = x 2 + - 8x + 30
y =
vertex =
Solution
  1. Factor out 1 from the x2 and x terms to get
    1(x2 - 8x) + 30
  2. Complete the square and subtract the extra to get
    1(x - 4)2 + 30 - 16
  3. Simplify to get the final answer.
#ProblemCorrect AnswerYour Answer
2y = x 2 + - 4x + 15
y =
vertex =
Solution
  1. Factor out 1 from the x2 and x terms to get
    1(x2 - 4x) + 15
  2. Complete the square and subtract the extra to get
    1(x - 2)2 + 15 - 4
  3. Simplify to get the final answer.

Complexity=3, Mode=hard

Complete the square for each equation and enter the vertex "(h, k)". It is possible that the equation is already in that form. Example:
"y = 3x2 + 12x + 7" becomes "y = 3(x + 2)2 - 5", with vertex (-2, -5).

#ProblemCorrect AnswerYour Answer
1y = 5x 2 + - 60x + 194
y =
vertex =
Solution
  1. Factor out 5 from the x2 and x terms to get
    5(x2 - 12x) + 194
  2. Complete the square and subtract the extra to get
    5(x - 6)2 + 194 - 180
  3. Simplify to get the final answer.
#ProblemCorrect AnswerYour Answer
2y = 4x 2 + - 72x + 327
y =
vertex =
Solution
  1. Factor out 4 from the x2 and x terms to get
    4(x2 - 18x) + 327
  2. Complete the square and subtract the extra to get
    4(x - 9)2 + 327 - 324
  3. Simplify to get the final answer.
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