Section 8.4: Quadratic Formula

The Quadratic Formula solves any quadratic equation in standard form \( ax^2 + bx + c = 0 \) using:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This method works even when factoring is difficult or impossible.

Example 1

Solve \( x^2 - 4x - 5 = 0 \) using the quadratic formula.

Step 1: Identify coefficients: \( a=1, b=-4, c=-5 \)

Step 2: Apply formula: \( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-5)}}{2(1)} \)

Step 3: Simplify: \( x = \frac{4 \pm \sqrt{16 + 20}}{2} = \frac{4 \pm \sqrt{36}}{2} \)

Step 4: Solve: \( x = \frac{4 \pm 6}{2} \) → \( x=5 \) or \( x=-1 \)

Solve \( 2x^2 + 3x - 2 = 0 \) using the quadratic formula.

Step 1: Identify coefficients: \( a=2, b=3, c=-2 \)

Step 2: Apply formula: \( x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} \)

Step 3: Simplify: \( x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4} \)

Step 4: Solve: \( x = \frac{-3+5}{4}= \frac{1}{2} \) or \( x = \frac{-3-5}{4} = -2 \)

Solve \( x^2 + 6x + 8 = 0 \)
Solve \( 3x^2 - x - 4 = 0 \)
Solve \( 5x^2 + 7x + 2 = 0 \)
Solve \( 2x^2 - 5x + 3 = 0 \)
Solve \( x^2 - x - 12 = 0 \)