Algebra (Higher tier) · Higher
Higher Algebra – Quadratics & Formula
Quadratic formula, completing the square, and solving equations with algebraic fractions – essential Higher skills.
Higher tier requires the quadratic formula and often completing the square. These methods work when factorisation is difficult or impossible.
The quadratic formula
For ax² + bx + c = 0:
x = [-b ± √(b² − 4ac)] / (2a)
Discriminant b² − 4ac:
• > 0 → two real solutions
• = 0 → one real solution (repeated root)
• < 0 → no real solutions
Worked example
Solve x² − 6x + 5 = 0 using the formula.
Solution
a = 1, b = −6, c = 5 x = [6 ± √(36 − 20)] / 2 = [6 ± √16] / 2 = [6 ± 4] / 2 **x = 5** or **x = 1**
Completing the square
For x² + bx + c:
x² + bx = (x + b/2)² − (b/2)²
Then rearrange to solve or to find turning point.
Worked example
Write x² + 8x + 10 in completed square form.
Solution
x² + 8x + 10 = (x + 4)² − 16 + 10 = **(x + 4)² − 6**
Practice questions
Try these without looking at the answers first.
Q1.Solve using the formula: x² + 5x + 6 = 0
Show answer
x = −2 or x = −3
Q2.Complete the square: x² − 6x + 2
Show answer
(x − 3)² − 7
Q3.Discriminant of x² − 4x + 7? How many real roots?
Show answer
−12, no real roots