Solving Quadratic Equations Using the Quadratic Formula

Introduction

• The quadratic formula provides a method to solve quadratic equations of the form: ( ax^2 + bx + c = 0 ).
• The formula is: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
• The goal is to find the values of ( x ) that satisfy the equation.

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

1. Identify coefficients:
   • ( a = 2 ), ( b = 3 ), ( c = -2 ).
2. Substitute into the quadratic formula: [ x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} ]
3. Simplify:
   • ( b^2 = 9 )
   • (-4ac = -4 \times 2 \times -2 = 16 )
   • ( 9 + 16 = 25 )
   • ( \sqrt{25} = 5 )
4. Calculate solutions:
   • ( x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} )
   • ( x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2 )
5. Verification:
   • Substitute ( x = -2 ) back into the original equation to confirm it equals zero.

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

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

Conclusion

• The quadratic formula is a useful tool for solving quadratic equations, providing exact solutions through substitution and simplification.
• Practice with different quadratics to become comfortable with the process.

Note

• Always check the solutions by substituting back into the original equation to ensure they satisfy it.