Quadratic Equations
6 minQuiz at the end
What is a Quadratic Equation?
A quadratic equation has the form:
axΒ² + bx + c = 0, where a β 0
The graph is a parabola. Solving a quadratic means finding the x-values where the parabola crosses the x-axis.
Method 1: Factorising
Find two numbers that multiply to c and add to b:
xΒ² + 5x + 6 = 0 β Numbers: 2 and 3 (2Γ3=6, 2+3=5) β (x + 2)(x + 3) = 0 β x = -2 or x = -3
Method 2: The Quadratic Formula
When factorising is difficult:
x = (-b Β± β(bΒ² - 4ac)) / 2a
For 2xΒ² - 3x - 2 = 0 (a=2, b=-3, c=-2):
- x = (3 Β± β(9+16)) / 4 = (3 Β± 5) / 4
- x = 8/4 = 2 or x = -2/4 = -0.5
Method 3: Completing the Square
Rewrite in the form (x + p)Β² + q = 0, then solve.
The Discriminant
Ξ = bΒ² - 4ac tells us how many real solutions exist:
| Ξ | Solutions |
|---|---|
| Ξ > 0 | Two distinct real solutions |
| Ξ = 0 | One repeated real solution |
| Ξ < 0 | No real solutions |