Solving Quadratic Equations Using the Quadratic Formula

Introduction

• Purpose: Learn how to solve quadratic equations using the quadratic formula.
• Example equation: (2x^2 + 3x - 2 = 0)

Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Components:
  • (a): coefficient of (x^2)
  • (b): coefficient of (x)
  • (c): constant term

Example 1

Given Equation

• (2x^2 + 3x - 2 = 0)
• Identify coefficients:
  • (a = 2)
  • (b = 3)
  • (c = -2)

Applying Quadratic Formula

1. Substitute into the formula:
   • (x = \frac{-3 \pm \sqrt{3^2 - 4 \times 2 \times -2}}{2 \times 2})
2. Calculate:
   • (x = \frac{-3 \pm \sqrt{9 + 16}}{4})
   • (x = \frac{-3 \pm 5}{4})
3. Solve for (x):
   • (x_1 = \frac{-3 + 5}{4} = \frac{1}{2})
   • (x_2 = \frac{-3 - 5}{4} = -2)

Verification

• Check solution by plugging (x = -2) back into the original equation.
• Calculation:
  • (2(-2)^2 + 3(-2) - 2 = 0)
  • Simplifies to (8 - 6 - 2 = 0), thus correct.

Example 2

Given Equation

• Coefficients:
  • (a = 6)
  • (b = -17)
  • (c = 12)

Applying Quadratic Formula

1. Substitute into the formula:
   • (x = \frac{17 \pm \sqrt{(-17)^2 - 4 \times 6 \times 12}}{2 \times 6})
2. Calculate:
   • (x = \frac{17 \pm \sqrt{289 - 288}}{12})
   • (x = \frac{17 \pm 1}{12})
3. Solve for (x):
   • (x_1 = \frac{17 + 1}{12} = \frac{18}{12} = \frac{3}{2})
   • (x_2 = \frac{17 - 1}{12} = \frac{16}{12} = \frac{4}{3})

Conclusion

• Successfully demonstrated how to apply the quadratic formula to solve quadratic equations.
• Provided verification for solutions to ensure accuracy.