Quadratic Equation Examples

\(x^2 - 9 = 0\) (Difference of Squares)

Step 1: Recognize \(x^2 - 3^2\)

Step 2: Factor: \((x - 3)(x + 3) = 0\)

Answer: \(x = 3\) or \(x = -3\)

\(x^2 + 5x + 6 = 0\)

Step 1: Find factors of 6 that sum to 5: 2 and 3

Step 2: Factor: \((x + 2)(x + 3) = 0\)

Answer: \(x = -2\) or \(x = -3\)

\(2x^2 - 5x - 3 = 0\)

Step 1: \(a \times c = 2 \times (-3) = -6\)

Step 2: Factors of -6 that sum to -5: -6 and 1

Step 3: Factor: \((2x + 1)(x - 3) = 0\)

Answer: \(x = -\frac{1}{2}\) or \(x = 3\)

\(x^2 - 4x + 2 = 0\)

a=1, b=-4, c=2

\(x = \frac{4 \pm \sqrt{16-8}}{2} = \frac{4 \pm \sqrt{8}}{2}\)

Answer: \(x = 2 \pm \sqrt{2}\)

\(3x^2 + 2x - 1 = 0\)

a=3, b=2, c=-1

\(x = \frac{-2 \pm \sqrt{4+12}}{6} = \frac{-2 \pm 4}{6}\)

Answer: \(x = \frac{1}{3}\) or \(x = -1\)

\(x^2 + x + 1 = 0\)

Discriminant: \(1 - 4 = -3\)

\(x = \frac{-1 \pm i\sqrt{3}}{2}\)

Answer: Complex roots

\(x^2 + 6x + 5 = 0\)

Step 1: \(x^2 + 6x = -5\)

Step 2: Add 9: \(x^2 + 6x + 9 = 4\)

Step 3: \((x+3)^2 = 4\)

Answer: \(x = -1\) or \(x = -5\)

\(2x^2 - 8x - 10 = 0\)

Step 1: Divide by 2: \(x^2 - 4x = 5\)

Step 2: Add 4: \((x-2)^2 = 9\)

Answer: \(x = 5\) or \(x = -1\)