Solving Quadratic Equations by Factoring

Introduction

• Topic: Solving quadratic equations by factoring.
• Importance of finding values of x that make the equation equal to zero.
• Process of solving quadratic equations includes factoring.

Example 1: Factoring x² - 5x = 0

• Equation: x² - 5x = 0
• Step 1: Identify greatest common factor (GCF).
  • GCF = x.
• Step 2: Factor the equation.
  • Factored form: x(x - 5) = 0.
• Step 3: Solve for x.
  • Factor 1: x = 0 (x₁).
  • Factor 2: x - 5 = 0 ⟹ x = 5 (x₂).
• Solutions: x₁ = 0, x₂ = 5.

Example 2: Factoring 4x² - 6x = 0

• Equation: 4x² - 6x = 0
• Step 1: GCF = 2x.
• Step 2: Factor the equation.
  • Factored form: 2x(2x - 3) = 0.
• Step 3: Solve for x.
  • Factor 1: 2x = 0 ⟹ x = 0 (x₁).
  • Factor 2: 2x - 3 = 0 ⟹ x = 3/2 (x₂).
• Solutions: x₁ = 0, x₂ = 3/2.

Example 3: Difference of Two Squares

• Equation: x² - 49 = 0
• Step 1: Recognize as difference of squares.
  • Factored form: (x + 7)(x - 7) = 0.
• Step 2: Solve for x.
  • Factor 1: x + 7 = 0 ⟹ x = -7 (x₁).
  • Factor 2: x - 7 = 0 ⟹ x = 7 (x₂).
• Solutions: x₁ = -7, x₂ = 7.

Example 4: Factoring 3x² - 75 = 0

• Equation: 3x² - 75 = 0
• Step 1: GCF = 3.
• Step 2: Factor the equation.
  • Factored form: 3(x² - 25) = 0 ⟹ 3(x + 5)(x - 5) = 0.
• Step 3: Solve for x.
  • Factor 1: x + 5 = 0 ⟹ x = -5 (x₁).
  • Factor 2: x - 5 = 0 ⟹ x = 5 (x₂).
• Solutions: x₁ = -5, x₂ = 5.

Example 5: Factoring a Trinomial

• Equation: x² - 2x - 15 = 0
• Step 1: Prepare two sets of parentheses.
• Step 2: Find factors of -15 that sum to -2.
  • Factors: -5 and 3.
• Step 3: Factor the equation.
  • Factored form: (x - 5)(x + 3) = 0.
• Step 4: Solve for x.
  • Factor 1: x - 5 = 0 ⟹ x = 5 (x₁).
  • Factor 2: x + 3 = 0 ⟹ x = -3 (x₂).
• Solutions: x₁ = 5, x₂ = -3.

Example 6: Assignment

• Problem: Factor x² + 5x + 6 = 0.

Example 7: More Factoring

• Equation: x² + 3x - 28 = 0
• Step 1: Prepare parentheses.
• Step 2: Find factors of -28 that sum to 3.
  • Factors: 4 and -7.
• Step 3: Factor the equation.
  • Factored form: (x + 7)(x - 4) = 0.
• Step 4: Solve for x.
  • Factor 1: x + 7 = 0 ⟹ x = -7 (x₁).
  • Factor 2: x - 4 = 0 ⟹ x = 4 (x₂).
• Solutions: x₁ = -7, x₂ = 4.

Conclusion

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