Completing the Square Worksheets Grade 10

Mathematics

Complete the square to solve quadratics and find vertex form

Each printable worksheet below is a full page of practice problems and comes with an answer key that explains how to solve every problem, step by step. Open a worksheet and use the Print / Save as PDF button to download it.

Worksheet 1

7 problems
  1. x² + 6x + 5 = 0 → (x + a)² + b = 0, find a and b
  2. x² + 6x + 8 = 0 → (x + h)² + k = 0; h = ?
  3. Complete the square: x² + 11x + 26 = 0 → (x + a)² = b, find a and b?

…and 4 more problems

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Worksheet 2

8 problems
  1. Charlotte is designing a parabolic arch for a new entrance to a botanical garden. The height of the arch (in meters) above the ground is given by the quadratic function h(x) = -2x^2 + 28x - 78, where x is the horizontal distance (in meters) from the left base of the arch. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to determine the maximum height of the arch and the horizontal distance from the left base where this maximum occurs. Then, state the coordinates of the vertex.
  2. Rewrite 2x² - 12x + 5 in the form a(x - h)² + k, and state the values of a, h, and k.
  3. Complete the square: 2x² - 16x + 10 = 0 → (x + a)² = b, find a and b?

…and 5 more problems

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Worksheet 3

6 problems
  1. Sophia is designing a parabolic arch for a new greenhouse entrance. The height of the arch (in meters) above the ground is given by the quadratic function h(x) = -4x^2 + 48x - 108, where x is the horizontal distance (in meters) from the left base of the arch. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to determine the maximum height of the arch and the horizontal distance from the left base where this maximum occurs. Then, state the coordinates of the vertex.
  2. Tane is designing a parabolic skateboard half-pipe for a community park. The cross-section of the ramp can be modeled by the quadratic function h(x) = -5x^2 + 50x - 45, where h(x) is the height in centimeters above the ground and x is the horizontal distance in centimeters from the left edge of the ramp. Using the method of completing the square, rewrite the function in vertex form h(x) = a(x - p)^2 + q to determine the maximum height of the ramp and the horizontal distance from the left edge where this maximum occurs. Then, state the coordinates of the vertex.
  3. Complete the square: x² + 14x + 29 = 0 → (x + a)² = b, find a and b?

…and 3 more problems

Open & Print Worksheet 3

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