Quadratic expressions and equations

Expanding

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

Special products:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b² (difference of two squares)

Factorising x² + bx + c

Find two numbers that multiply to c and add to b.

x² + 7x + 12: numbers 3 and 4 (3 x 4 = 12, 3 + 4 = 7) → (x + 3)(x + 4)

x² - x - 6: numbers -3 and 2 → (x - 3)(x + 2)

x² - 25 = (x - 5)(x + 5)

Solving quadratic equations by factorising

x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3

The quadratic formula

For ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a

2x² + 3x - 2 = 0: x = (-3 ± √(9 + 16))/4 = (-3 ± 5)/4 → x = 0.5 or x = -2