Solving Quadratic Equations Using the Quadratic Formula

Introduction

• Focus on using the quadratic formula to solve quadratic equations.

Quadratic Formula

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

Example 1: Solving ( 2x^2 + 3x - 2 = 0 )

1. Identify coefficients:
   • ( a = 2 ), ( b = 3 ), ( c = -2 )
2. Apply quadratic formula: [ x = \frac{-3 \pm \sqrt{3^2 - 4 \times 2 \times (-2)}}{2 \times 2} ]
3. Simplify:
   • ( x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4} )
   • Two solutions: ( x = \frac{2}{4} = \frac{1}{2} ) and ( x = \frac{-8}{4} = -2 )
4. Verification:
   • Plug ( x = -2 ) back into the equation:
     • ( 2(-2)^2 + 3(-2) - 2 = 0 )
     • Simplifies to ( 0 = 0 ), hence correct.

Example 2: Solving ( 6x^2 - 17x + 12 = 0 )

1. Identify coefficients:
   • ( a = 6 ), ( b = -17 ), ( c = 12 )
2. Apply quadratic formula: [ x = \frac{17 \pm \sqrt{(-17)^2 - 4 \times 6 \times 12}}{2 \times 6} ]
3. Simplify:
   • ( x = \frac{17 \pm \sqrt{289 - 288}}{12} )
   • ( x = \frac{17 \pm 1}{12} )
   • Two solutions: ( x = \frac{18}{12} = \frac{3}{2} ) and ( x = \frac{16}{12} = \frac{4}{3} )

Conclusion

• The quadratic formula provides a reliable method to find the roots of quadratic equations.
• Practice with various equations to strengthen understanding.

Thanks for watching the tutorial on using the quadratic formula!