Quadratic equations
- A quadratic equation can be written as ax² + bx + c = 0. It can have two solutions (roots), one repeated solution, or no real solutions.
- Always rearrange so one side is 0 before solving.
Solving by factorising
- Factorise, then set each bracket equal to 0: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = −2 or x = −3.
- x² = 25 gives x = 5 or x = −5. Don't forget the negative root.
- x² − 7x = 0 → x(x − 7) = 0 → x = 0 or x = 7. Never divide both sides by x: you lose the solution x = 0.
- (Higher) 2x² + 7x + 3 = 0 → (2x + 1)(x + 3) = 0 → x = −½ or x = −3.
The quadratic formula (Higher)
- x = (−b ± √(b² − 4ac)) / 2a. Use it when the quadratic won't factorise, and give answers to the accuracy asked.
- 2x² − 3x − 7 = 0: a = 2, b = −3, c = −7, so x = (3 ± √65) / 4, giving x = 2.77 or x = −1.27 (2 d.p.).
- The value of b² − 4ac tells you how many solutions there are: positive gives two, zero gives one, negative gives none.
Completing the square (Higher)
- x² + bx + c = (x + b/2)² − (b/2)² + c. For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
- The turning point of y = (x + 3)² − 7 is (−3, −7).
- To solve (x + 3)² − 7 = 0: (x + 3)² = 7, so x = −3 ± √7.
Key terms
- Quadratic equation
- An equation where the highest power of x is 2.
- Root
- A solution of an equation; where a graph crosses the x-axis.
- Quadratic formula
- x = (−b ± √(b² − 4ac)) / 2a, which solves any quadratic.
- Completing the square
- Writing a quadratic in the form (x + p)² + q.
- Turning point
- The minimum or maximum point of a quadratic graph.