Quadratic Completing Square

Grade 9 · algebra · 76 practice problems · read aloud

🔊 Listen to this explanation

What is Completing the Square? ✨

Completing the Square is a method that transforms a quadratic equation into a perfect square trinomial. This helps us:

  • Solve quadratic equations (find x-intercepts)
  • Find the vertex of a parabola (maximum/minimum point)
  • Rewrite quadratic functions in vertex form: y = a(x - h)² + k

Step-by-Step Guide 📝

  1. Start with: ax² + bx + c = 0
  2. Move constant: ax² + bx = -c
  3. Divide by a (if a ≠ 1): x² + (b/a)x = -c/a
  4. Find magic number: (b/2)²
  5. Add to both sides: x² + bx + (b/2)² = -c + (b/2)²
  6. Factor left side: (x + b/2)² = right side
  7. Solve for x: Take square root of both sides

Worked Examples 🧮

Example 1: x² + 6x + 5 = 0

Step 1: Move constant: x² + 6x = -5

Step 2: Find (6/2)² = 9

Step 3: Add to both sides: x² + 6x + 9 = -5 + 9

Step 4: Factor: (x + 3)² = 4

Step 5: Solve: x + 3 = ±2 → x = -1 or x = -5

Example 2: 2x² - 8x + 1 = 0

Step 1: Divide by 2: x² - 4x + 0.5 = 0

Step 2: Move constant: x² - 4x = -0.5

Step 3: Find (-4/2)² = 4

Step 4: Add to both sides: x² - 4x + 4 = -0.5 + 4

Step 5: Factor: (x - 2)² = 3.5

Step 6: Solve: x - 2 = ±√3.5 → x = 2 ± √3.5

⚠️ Common Mistakes to Avoid

  • Forgetting to divide by a first when a ≠ 1
  • Adding (b/2)² to only one side of the equation
  • Wrong sign in the factored form: (x + b/2)² not (x - b/2)²
  • Missing ± when taking square roots

💡 Tips & Tricks

  • Memory aid: "Half the middle, square it, add to both"
  • Check your work: Expand (x + h)² to verify
  • Vertex connection: After (x - h)² + k, vertex is at (h, k)
  • Practice pattern: The middle term is always 2 times your h value

Practice Suggestions 📚

  • Start with simple quadratics where a = 1
  • Practice identifying perfect square trinomials
  • Work backwards: expand (x + h)² to see the pattern
  • Mix with factoring and quadratic formula to see connections
  • Use graph to check: vertex should match your (h,k) values

Practice problems

6 of the 76, worked through step by step — try them before opening the answer.

1 x² + 6x - 7 = 0

Hint: To solve a quadratic equation by completing the square, first move the constant term to the right side, then add the square of half the coefficient of x to both sides to create a perfect square trinomial on the left.

Show the answer

Answer: x = 1, x = -7

  1. Identify the coefficients. The equation is in the form: a x² + b x + c = 0 Here, a = 1, b = 6, c = -7.
  2. Factor the quadratic expression. We look for two numbers that multiply to a * c = 1 * (-7) = -7 and add to b = 6. Possible pairs for -7: 1 and -7 → multiply to -7, add to -6 (no) -1 and 7 → multiply to -7, add to 6 (yes) So the numbers are -1 and 7.
  3. Rewrite the middle term using these numbers. x² + 6x - 7 = x² - 1x + 7x - 7
  4. Factor by grouping. Group terms: (x² - 1x) + (7x - 7) Factor each group: x(x - 1) + 7(x - 1)
  5. Factor out the common factor (x - 1). (x - 1)(x + 7) = 0
  6. Apply the zero-product property. If (x - 1)(x + 7) = 0, then: x - 1 = 0 or x + 7 = 0
  7. Solve each equation. x - 1 = 0 → x = 1 x + 7 = 0 → x = -7 Final answer: x = 1, x = -7

We are solving the quadratic equation: x² + 6x - 7 = 0

2 x² - 8x + 5 = 0

Hint: Move the constant term to the right side, then add the square of half the coefficient of x to both sides to create a perfect square trinomial.

Show the answer

Answer: 4 ± √11

  1. Move the constant term to the right side: x² - 8x = -5
  2. Take half of the coefficient of x: (-8)/2 = -4
  3. Square this result: (-4)² = 16
  4. Add 16 to both sides: x² - 8x + 16 = -5 + 16
  5. Simplify the right side: x² - 8x + 16 = 11
  6. Factor the left side as a perfect square: (x - 4)² = 11
  7. Take the square root of both sides: x - 4 = ±√11
  8. Add 4 to both sides: x = 4 ± √11

The answer is 4 ± √11.

3 x² + 6x - 4 = 0

Hint: Move the constant term to the other side, then find the number that completes the square when added to both sides.

Show the answer

Answer: -3 ± √13

  1. Move the constant term to the right side: x² + 6x = 4
  2. Find the number that completes the square: (6/2)² = 9
  3. Add 9 to both sides: x² + 6x + 9 = 4 + 9
  4. Factor the perfect square trinomial: (x + 3)² = 13
  5. Take the square root of both sides: x + 3 = ±√13
  6. Solve for x: x = -3 ± √13

The answer is -3 ± √13.

4 x² - 8x + 13 = 0

Hint: Move the constant term to the other side, then add a specific number to both sides to create a perfect square trinomial on the left.

Show the answer

Answer: 4 ± √3

  1. Move the constant term to the right side: x² - 8x = -13
  2. Find the number to complete the square: take half of -8, which is -4, then square it to get 16
  3. Add 16 to both sides: x² - 8x + 16 = -13 + 16
  4. Write the left side as a perfect square: (x - 4)² = 3
  5. Take the square root of both sides: x - 4 = ±√3
  6. Solve for x: x = 4 ± √3
  7. The solutions are x = 4 + √3 and x = 4 - √3

5 x² - 10x + 18 = 0

Hint: For a quadratic equation like y² - 6y + 4 = 0, you can complete the square by finding what number makes the left side a perfect square trinomial.

Show the answer

Answer: 5 ± √7

  1. Move the constant term to the right side: x² - 10x = -18
  2. Find the number to complete the square: (10/2)² = 25
  3. Add 25 to both sides: x² - 10x + 25 = -18 + 25
  4. Simplify: x² - 10x + 25 = 7
  5. Factor the perfect square trinomial: (x - 5)² = 7
  6. Take the square root of both sides: x - 5 = ±√7
  7. Solve for x: x = 5 ± √7

The answer is 5 ± √7.

6 x² + 10x + 18 = 0

Hint: Move the constant term to the right side, then find the number that completes the perfect square trinomial when added to both sides.

Show the answer

Answer: -5 ± √7

  1. Move the constant term to the right side: x² + 10x = -18
  2. Find the number to complete the square: (10/2)² = 25
  3. Add 25 to both sides: x² + 10x + 25 = -18 + 25
  4. Factor the perfect square trinomial: (x + 5)² = 7
  5. Take square root of both sides: x + 5 = ±√7
  6. Solve for x: x = -5 ± √7

The answer is -5 ± √7.

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