Solving Quadratics Using the Quadratic Formula

Introduction

• Quadratics can be solved via multiple methods:
  • Graphical representation
  • Factorization
  • Trial and improvement
  • Quadratic Formula
• Quadratic formula is particularly useful in exams and complex scenarios.
• Formula to be memorized at higher levels of education.

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: Solving ( x^2 + 8x + 15 = 0 )

1. Identify coefficients
   • ( a = 1 ) (coefficient of ( x^2 ))
   • ( b = 8 ) (coefficient of ( x ))
   • ( c = 15 ) (constant term)
2. Substitute into formula
   • ( x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} )
3. Simplify
   • ( x = \frac{-8 \pm \sqrt{64 - 60}}{2} )
   • ( x = \frac{-8 \pm \sqrt{4}}{2} )
   • Solutions: ( x = -3 ) or ( x = -5 )

Example 2: Solving ( x^2 - 10x - 5 = 0 ) to Two Decimal Places

1. Identify coefficients
   • ( a = 1 )
   • ( b = -10 )
   • ( c = -5 )
2. Substitute into formula
   • ( x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} )
   • ( x = \frac{10 \pm \sqrt{100 + 20}}{2} )
3. Simplify and Calculate
   • ( x = \frac{10 \pm \sqrt{120}}{2} )
   • Use calculator for final values:
     • ( x \approx 10.48 ) to 2 decimal places
     • ( x \approx -0.48 ) to 2 decimal places

Tips

• Be cautious with signs, especially with negative numbers.
• Always check if factorization is possible for quicker solutions.
• Quadratic formula is essential for non-factorizable quadratics and precise decimal solutions.