Knowledge Organiser: Solving Quadratic Equations
Key Terms
- Quadratic equation: An equation of the form ax² + bx + c = 0
- Root / solution: The value(s) of x that satisfy the equation
- Null factor law: If A × B = 0, then A = 0 or B = 0
- Parabola: The U-shaped graph of a quadratic: roots are where it crosses the x-axis
- Repeated root: When both solutions are the same (parabola touches x-axis)
Must-Know Facts
- ALWAYS rearrange to ax² + bx + c = 0 before factorising
- A quadratic can have 0, 1, or 2 solutions
- From (x + a)(x + b) = 0: solutions are x = −a and x = −b
- If x² = k, then x = +√k or x = −√k
- For x(x + 3) = 0: solutions are x = 0 and x = −3
- Check both solutions by substituting back into the original equation
Key Methods
- Factorising: rearrange to = 0, factorise, set each bracket = 0
- Difference of two squares: x² − k² = 0 → x = ±k
- Simple form: ax² + c = 0 → x² = −c/a → x = ±√(−c/a)
- Higher tier only: use the quadratic formula when factorising is not possible
Common Mistakes
- Not rearranging to = 0 first: You must have ax² + bx + c = 0 before factorising: do not try to factorise with a constant on the right-hand side
- Wrong signs from brackets: From (x + 2)(x − 5) = 0, solutions are x = −2 and x = 5: the solutions are the opposites of the numbers in the brackets
- Forgetting the negative root: If x² = 9, then x = 3 OR x = −3: always give both roots unless the context rules one out
- Cancelling x from x² = 3x: Divide only after moving everything to one side: x² − 3x = 0 → x(x − 3) = 0, giving x = 0 and x = 3