Completing the Square

Grade 10 · mathematics · 79 practice problems · read aloud

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Completing the Square

What is it and Why Use It?

Completing the Square is an algebraic technique that rewrites a quadratic expression from standard form (ax²+bx+c) into vertex form (a(x-h)²+k). This is incredibly useful for:

  • Finding the vertex of a parabola 📈
  • Graphing quadratic functions easily
  • Solving quadratic equations
  • Deriving the Quadratic Formula itself!

Step-by-Step Guide

  1. Step 1: Start with a quadratic in standard form: ax² + bx + c = 0.
  2. Step 2: If a ≠ 1, divide the entire equation by 'a'.
  3. Step 3: Move the constant term (c) to the right side of the equation.
  4. Step 4: Take half of the b-coefficient, square it, and add it to BOTH sides.
  5. Step 5: Factor the perfect square trinomial on the left side.
  6. Step 6: Solve for x by taking the square root of both sides.

Worked Examples

Example 1: x² + 6x + 5 = 0

  1. Move constant: x² + 6x = -5
  2. Half of 6 is 3, 3² = 9. Add 9 to both sides: x² + 6x + 9 = -5 + 9
  3. Factor: (x + 3)² = 4
  4. Solve: x + 3 = ±2 → x = -1 or x = -5

Example 2: 2x² - 8x - 10 = 0

  1. Divide by 2: x² - 4x - 5 = 0
  2. Move constant: x² - 4x = 5
  3. Half of -4 is -2, (-2)² = 4. Add 4 to both sides: x² - 4x + 4 = 5 + 4
  4. Factor: (x - 2)² = 9
  5. Solve: x - 2 = ±3 → x = 5 or x = -1

Common Mistakes to Avoid

  • Forgetting to add the constant to both sides. If you add it to one side, you MUST add it to the other.
  • Using the wrong sign in the factored form. The sign inside (x-h)² comes from the b-term's sign in x²+bx.
  • Not dividing all terms by 'a' when a ≠ 1 at the very beginning.
  • Mishandling the ± sign when taking the square root.

Tips & Tricks

  • Memory Aid: "Half it, square it, add it to both."
  • The number you add is always (b/2)².
  • The vertex of the parabola is at (h, k) from the form a(x-h)²+k.
  • Check your work by expanding your final factored form.

How to Practice

Start simple! Practice with quadratics where a=1 and b is an even number. Then move to cases where a≠1. Try these:

  • x² + 8x + 7 = 0
  • x² - 2x - 15 = 0
  • 3x² + 12x - 15 = 0

Convert them to vertex form and find the vertex. The more you do, the more natural it becomes!

Practice problems

6 of the 79, worked through step by step — try them before opening the answer.

1 x² + 6x - 7 = 0 → (x + a)² = b

Hint: To complete the square, take half of the coefficient of the linear term, square it, and add/subtract appropriately to form a perfect square trinomial.

Show the answer

Answer: a=3, b=16

  1. Move the constant term to the other side** x^2 + 6x = 7 --- **
  2. Complete the square** Take the coefficient of x, which is 6. Divide it by 2: 6/2 = 3 Square it: 3^2 = 9 Add 9 to both sides of the equation: x^2 + 6x + 9 = 7 + 9 --- **
  3. Factor the left-hand side** The left-hand side is now a perfect square trinomial: (x + 3)^2 = 16 --- **
  4. Compare with (x + a)^2 = b** We have (x + 3)^2 = 16 So a = 3 and b = 16 --- **Final answer:** a = 3, b = 16

We start with the equation: x^2 + 6x - 7 = 0 We want to rewrite it in the form: (x + a)^2 = b --- **

2 x² - 8x + 13 = 0 → (x - h)² = k, find k?

Hint: To complete the square, take half of the coefficient of x, square it, and adjust the constant term accordingly. For example, in x² + 4x + c, half of 4 is 2, and 2 squared is 4, so (x + 2)² = x² + 4x + 4.

Show the answer

Answer: 3

  1. Start with x² - 8x + 13 = 0
  2. Move the constant term to the other side: x² - 8x = -13
  3. Take half of the coefficient of x: (-8)/2 = -4
  4. Square this result: (-4)² = 16
  5. Add 16 to both sides: x² - 8x + 16 = -13 + 16
  6. The left side becomes a perfect square: (x - 4)² = 3
  7. Compare with (x - h)² = k, so k = 3

The answer is 3.

3 x² + 6x + 8 = 0 → (x + h)² + k = 0; h = ?

Hint: To complete the square for a quadratic expression, take half of the coefficient of the linear term and square it to find the constant needed to form a perfect square trinomial.

Show the answer

Answer: 3

  1. Understand the goal** The expression (x + h)^2 expands to: x^2 + 2h x + h^2 So we need to match the coefficients of x^2 + 6x + 8 with x^2 + 2h x + h^2 + k. --- **
  2. Compare coefficients of x** From x^2 + 6x + 8: Coefficient of x is 6. From expansion: coefficient of x is 2h. So: 2h = 6 h = 3 --- **
  3. Check constant term** From expansion: (x + h)^2 + k = x^2 + 2h x + h^2 + k. Constant term in original equation: 8. So: h^2 + k = 8. Since h = 3: 9 + k = 8 k = -1 --- **
  4. Verify** (x + 3)^2 - 1 = x^2 + 6x + 9 - 1 = x^2 + 6x + 8. Matches the original equation. --- **Final answer:** h = 3

We start with the equation: x^2 + 6x + 8 = 0 We want to rewrite it in the form: (x + h)^2 + k = 0 --- **

4 x² + 6x - 7 = 0 → (x + a)² = b, find a and b

Hint: To complete the square, take half of the coefficient of the linear term, square it, and add it to both sides of the equation. This creates a perfect square trinomial on one side.

Show the answer

Answer: a=3, b=16

  1. Move the constant term to the other side** x^2 + 6x = 7 --- **
  2. Complete the square** Take the coefficient of x, which is 6. Divide it by 2: 6/2 = 3 Square it: 3^2 = 9 Add 9 to both sides of the equation: x^2 + 6x + 9 = 7 + 9 --- **
  3. Write the left side as a perfect square** x^2 + 6x + 9 = (x + 3)^2 So: (x + 3)^2 = 16 --- **
  4. Compare with (x + a)^2 = b** We have: a = 3 b = 16 --- **Final answer:** a = 3, b = 16

We start with the equation: x^2 + 6x - 7 = 0 We want to rewrite it in the form: (x + a)^2 = b --- **

5 x² + 8x + 13 = 0 → (x + a)² = b, find a and b?

Hint: To complete the square, take half of the coefficient of x, square it, and adjust the constant term accordingly.

Show the answer

Answer: a=4, b=3

  1. Start with x² + 8x + 13 = 0
  2. Move constant term: x² + 8x = -13
  3. Take half of 8: 8/2 = 4
  4. Square it: 4² = 16
  5. Add 16 to both sides: x² + 8x + 16 = -13 + 16
  6. Simplify: x² + 8x + 16 = 3
  7. Factor perfect square: (x + 4)² = 3
  8. Therefore, a = 4 and b = 3

6 x² - 14x + 47 = 0 → (x - h)² = k, find h and k?

Hint: To rewrite a quadratic in the form (x - h)² = k, you need to create a perfect square trinomial. Focus on the coefficient of x: take half of it, square that value, and then adjust the constant term to maintain equality.

Show the answer

Answer: h=7, k=2

  1. Start with x² - 14x + 47 = 0
  2. Move the constant term to the right side: x² - 14x = -47
  3. Take half of the coefficient of x: -14/2 = -7
  4. Square this value: (-7)² = 49
  5. Add 49 to both sides: x² - 14x + 49 = -47 + 49
  6. Simplify the right side: -47 + 49 = 2
  7. The left side is now a perfect square trinomial: (x - 7)² = 2
  8. Therefore, h = 7 and k = 2.
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