Section 8.5: Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form \( (x+p)^2 = q \).

General steps:

  1. Move constant term to the other side.
  2. Divide coefficient of \(x^2\) if not 1.
  3. Add \((\frac{b}{2})^2\) to both sides.
  4. Factor the perfect square trinomial.
  5. Take the square root and solve for \(x\).

Example 1

Solve \( x^2 + 6x - 7 = 0 \) by completing the square.

Step 1: Move constant: \( x^2 + 6x = 7 \)

Step 2: Compute \((\frac{6}{2})^2 = 9\), add to both sides: \( x^2 + 6x + 9 = 7 + 9 \)

Step 3: Factor left side: \( (x + 3)^2 = 16 \)

Step 4: Solve: \( x + 3 = \pm 4 \) → \( x = 1 \) or \( x = -7 \)

Example 2

Solve \( 2x^2 + 8x - 10 = 0 \) by completing the square.

Step 1: Divide by 2: \( x^2 + 4x - 5 = 0 \) → \( x^2 + 4x = 5 \)

Step 2: Compute \((\frac{4}{2})^2 = 4\), add to both sides: \( x^2 + 4x + 4 = 9 \)

Step 3: Factor left side: \( (x + 2)^2 = 9 \)

Step 4: Solve: \( x + 2 = \pm 3 \) → \( x = 1 \) or \( x = -5 \)

Practice Problems

  1. Solve \( x^2 + 10x + 21 = 0 \)
  2. Solve \( x^2 - 4x - 5 = 0 \)
  3. Solve \( 3x^2 + 12x - 15 = 0 \)
  4. Solve \( x^2 + 8x + 12 = 0 \)
  5. Solve \( 2x^2 - 4x - 6 = 0 \)
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