Solving Quadratic Equations with the Quadratic Formula

Introduction

• Learn to use the quadratic formula to solve quadratic equations.
• Solutions should be given to two decimal places.

Quadratic Formula

• Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
• Important: Memorize this formula for exams as it won't be provided.

Components of the Quadratic Formula

• Standard form of a quadratic equation: ( ax^2 + bx + c = 0 )
  • ( a ): Coefficient of ( x^2 )
  • ( b ): Coefficient of ( x )
  • ( c ): Constant term

Example 1: Solving a Quadratic Equation

Given Equation

• ( 2x^2 + 5x + 1 = 0 )

Steps to Solve

1. Identify ( a ), ( b ), and ( c ):
   • ( a = 2 ), ( b = 5 ), ( c = 1 )
2. Substitute into the formula:
   • Substitute values into ( x = \frac{-5 \pm \sqrt{5^2 - 4 \times 2 \times 1}}{2 \times 2} )
3. Simplify inside the square root:
   • ( x = \frac{-5 \pm \sqrt{25 - 8}}{4} )
   • ( x = \frac{-5 \pm \sqrt{17}}{4} )
4. Calculate the solutions:
   • ( x_1 = \frac{-5 + \sqrt{17}}{4} )
   • ( x_2 = \frac{-5 - \sqrt{17}}{4} )
   • Approximate: ( x_1 = -0.22 ), ( x_2 = -2.28 )

Example 2: Solving a More Complex Quadratic Equation

Given Equation

• ( x^2 - 5x + 7 = 10 - 2x^2 )

Steps to Solve

1. Rearrange to standard form:
   • Move terms to get ( 3x^2 - 5x - 3 = 0 )
2. Identify ( a ), ( b ), and ( c ):
   • ( a = 3 ), ( b = -5 ), ( c = -3 )
3. Substitute into the formula:
   • ( x = \frac{5 \pm \sqrt{(-5)^2 - 4 \times 3 \times (-3)}}{2 \times 3} )
4. Simplify inside the square root:
   • ( x = \frac{5 \pm \sqrt{25 + 36}}{6} )
   • ( x = \frac{5 \pm \sqrt{61}}{6} )
5. Calculate the solutions:
   • ( x_1 = \frac{5 + \sqrt{61}}{6} )
   • ( x_2 = \frac{5 - \sqrt{61}}{6} )
   • Approximate: ( x_1 = 2.14 ), ( x_2 = -0.468 )

Conclusion

• Use the quadratic formula for non-factorable quadratics.
• Check calculations with a calculator, especially for decimal or complex roots.
• Practice substituting and simplifying to become efficient.

Recommendations

• Practice with more equations to master the use of the quadratic formula.
• Subscribe for future educational content.