Learning Objectives
- Identify and represent quadratic equations in standard form
- Solve quadratic equations by factorisation
- Solve quadratic equations using the quadratic formula
- Determine the nature of roots using the discriminant
Key Concepts
Standard Form
A quadratic equation in variable x is of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Solution by Factorisation
Write the middle term as a sum of two terms whose product equals ac. Factor by grouping and apply the zero-product property: if (x - α)(x - β) = 0, then x = α or x = β.
Example: x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3
Completing the Square
Transform ax² + bx + c = 0 into the form (x + p)² = q², then solve by taking square roots.
Quadratic Formula
The roots of ax² + bx + c = 0 are given by:
x = (-b ± √(b² - 4ac)) / 2a
Nature of Roots (Discriminant D = b² - 4ac)
- D > 0: Two distinct real roots
- D = 0: Two equal real roots (repeated root x = -b/2a)
- D < 0: No real roots (roots are complex/imaginary)
Summary
Quadratic equations are polynomial equations of degree 2. They can be solved by factorisation, completing the square, or the quadratic formula. The discriminant (b² - 4ac) determines the nature and number of real roots. These equations model many real-world problems involving areas, projectile motion, and profit/loss.
Important Terms
- Quadratic Equation
- An equation of the form ax² + bx + c = 0 where a ≠ 0
- Roots
- The values of x that satisfy the quadratic equation
- Discriminant
- D = b² - 4ac; determines the nature of roots
- Zero-Product Property
- If ab = 0, then a = 0 or b = 0
Quick Revision
- Standard form: ax² + bx + c = 0 (a ≠ 0)
- Quadratic formula: x = [-b ± √(b²-4ac)] / 2a
- D > 0: two distinct real roots; D = 0: two equal roots; D < 0: no real roots
- Sum of roots = -b/a; Product of roots = c/a
- Always check solutions by substituting back into the original equation