Quadratic Equations
What You Need to Know
- A quadratic equation is any equation in the form ax² + bx + c = 0
- Three main methods: factorising, completing the square, quadratic formula
- You must be able to identify which method to use for a given equation
- Quadratic graphs are parabolas — know how to sketch them
Method 1: Factorising
When a quadratic can be written as (x + p)(x + q) = 0, you can solve by setting each bracket to zero. This only works when the quadratic 'factors nicely'.
Worked Example
Solve x² + 5x + 6 = 0
Find two numbers that multiply to 6 and add to 5 → 2 and 3Write as (x + 2)(x + 3) = 0Set each bracket to zero: x + 2 = 0 or x + 3 = 0Solutions: x = -2 or x = -3
Answer: x = -2 or x = -3
Method 2: Quadratic Formula
The formula x = (-b ± √(b²-4ac)) / 2a works for ALL quadratic equations. Use this when factorising doesn't work.
Worked Example
Solve 2x² + 3x - 5 = 0
Identify: a = 2, b = 3, c = -5Substitute into formula: x = (-3 ± √(9 + 40)) / 4Simplify: x = (-3 ± √49) / 4 = (-3 ± 7) / 4Two solutions: x = (-3 + 7)/4 = 1 or x = (-3 - 7)/4 = -2.5
Answer: x = 1 or x = -2.5
Method 3: Completing the Square
Rewrite the quadratic in the form (x + p)² = q. Useful for finding turning points and when asked specifically to use this method.
Worked Example
Solve x² + 6x + 2 = 0 by completing the square
Take half the coefficient of x: 6/2 = 3Write as (x + 3)² - 9 + 2 = 0Simplify: (x + 3)² = 7Square root both sides: x + 3 = ±√7Solutions: x = -3 + √7 or x = -3 - √7
Answer: x = -3 + √7 or x = -3 - √7
Common Mistakes
- Forgetting the ± when using the square root — quadratics usually have TWO solutions
- Sign errors when factorising (x - 3)(x + 5) gives roots x = 3 and x = -5, not the other way around
- Using the wrong a, b, c values — always rearrange to ax² + bx + c = 0 FIRST
- Trying to factorise when the discriminant (b² - 4ac) is negative — use the formula instead
Exam Tips
- AO1: Usually 2-3 marks for solving — show your working clearly
- AO2: May ask you to form a quadratic from a word problem first
- AO3: Could involve proving something about quadratics or interpreting solutions in context
Related Topics
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