Quadratic Equations: Methods, Discriminant and Word Problems
Quadratic Equations appears as a 1-mark discriminant question, a 2–3 mark solution by factorisation or formula, and a 3–4 mark word problem on speed, area or numbers.
Methods and the discriminant
Bring the equation to ax² + bx + c = 0, a ≠ 0.
- Factorisation: split b into two numbers whose product is ac and whose sum is b.
- Completing the square: build (x + b/2a)².
- Formula: x = [−b ± √(b² − 4ac)]/2a.
Discriminant D = b² − 4ac:
- D > 0: two distinct real roots.
- D = 0: two equal real roots, each −b/2a.
- D < 0: no real roots.
Roots also satisfy α + β = −b/a and αβ = c/a.
Worked example
Solve 2x² − 7x + 3 = 0. D = (−7)² − 4(2)(3) = 49 − 24 = 25 > 0, so there are two distinct real roots. x = (7 ± 5)/4, giving x = 3 or x = 1/2.
Word problems
Form the equation from the statement, solve, then reject roots that are physically impossible (negative length, negative time, unrealistic age). Answer the quantity asked for - often a speed or a side, not x itself.
Traps that cost marks
- D < 0 means no real roots, not "no solution"; say real explicitly.
- The formula must carry the signs of b and c: −(−7) = +7.
- For "equal roots" questions set D = 0 and solve for the unknown constant; do not solve the quadratic.
- Never cancel a variable across an equation (x² = 5x → x = 5 loses x = 0); factorise instead.
- Final answers need units and a statement.