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Solving Quadratic Equations by Completing the Square

Jul 13, 2024

Solving Quadratic Equations by Completing the Square

General Form of Quadratic Equation

  • The general form is: x² + bx + c = 0 where a = 1.

Method Overview

  1. Ensure the quadratic equation is in the general form with a = 1.
  2. Move the constant term c to the other side of the equation.
  3. Add a number to both sides to complete the square.
  4. Factorize the left-hand side into a squared term.
  5. Solve for x by isolating the variable.
  6. Provide answers to three decimal places.

Example Problems

Question A

Equation: x² + 4x - 9 = 0

  1. Move -9: x² + 4x = 9
  2. (4/2)² = 4, add to both sides: x² + 4x + 4 = 9 + 4
  3. Factorize: (x + 2)² = 13
  4. Solve: x + 2 = ±√13
  5. Solutions: x = -2 ± √13
  6. Decimal solutions: x ≈ 1.606, x ≈ -5.606

Question B

Equation: x² - 3x - 5 = 0

  1. Move -5: x² - 3x = 5
  2. (-3/2)² = 9/4, add to both sides: x² - 3x + 9/4 = 5 + 9/4
  3. Factorize: (x - 3/2)² = 29/4
  4. Solve: x - 3/2 = ±√(29/4)
  5. Solutions: x = 3/2 ± √(29/4)
  6. Decimal solutions: x ≈ 4.366, x ≈ -1.193

Question C

Equation: -x² - 6x + 9 = 0

  1. Divide by -1: x² + 6x - 9 = 0
  2. Move -9: x² + 6x = 9
  3. (6/2)² = 9, add to both sides: x² + 6x + 9 = 18
  4. Factorize: (x + 3)² = 18
  5. Solve: x + 3 = ±√18
  6. Solutions: x = -3 ± √18
  7. Decimal solutions: x ≈ 1.243, x ≈ -7.243

Question D

Equation: 2x² - 6x + 3 = 0

  1. Divide by 2: x² - 3x + 3/2 = 0
  2. Move 3/2: x² - 3x = -3/2
  3. (-3/2/2)² = 9/16, add to both sides: x² - 3x + 9/16 = 0
  4. Factorize: (x - 3/2)² = -3/2 + 9/16
  5. Solve: x - 3/2 = ±√(3/4)
  6. Solutions: x = 3/2 ± √(3/4)
  7. Decimal solutions: x ≈ 2.366, x ≈ 0.634

Question E

Equation: 4x² - 8x + 1 = 0

  1. Divide by 4: x² - 2x + 1/4 = 0
  2. Move 1/4: x² - 2x = -1/4
  3. (-2/2)² = 1, add to both sides: x² - 2x + 1 = 3/4
  4. Factorize: (x - 1)² = -1/4 + 3/4
  5. Solve: x - 1 = ±√(3/4)
  6. Solutions: x = 1 ± √(3/4)
  7. Decimal solutions: x ≈ 1.866, x ≈ 0.134

Question F

Equation: -2x² + 7x + 6 = 0

  1. Divide by -2: x² - (7/2)x - 3 = 0
  2. Move -3: x² - (7/2)x = 3
  3. (-7/4)² = 49/16, add to both sides: x² - (7/4)x + 49/16 = 97/16
  4. Factorize: (x - 7/4)² = 97/16
  5. Solve: x - 7/4 = ±√(97/16)
  6. Solutions: x = 7/4 ± √(97/16)
  7. Decimal solutions: x ≈ 4.212, x ≈ -0.712

Summary

  • Completing the square converts a quadratic equation into a form that can be easily solved by taking square roots.
  • The steps involve balancing the equation and creating perfect square trinomials.
  • Providing answers to three decimal places ensures precision in the results.

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