✔ Official Ocoviz resource · created by the Ocoviz team, free for every student
Factorisation, completing the square, and the quadratic formula — when to use each, with worked examples and the discriminant shortcut.
A quadratic equation is any equation that can be written as ax^2 + bx + c = 0, where a is not 0.
METHOD 1 — FACTORISATION (fastest when it works)
Split the middle term b into two numbers that MULTIPLY to a*c and ADD to b.
Example: x^2 + 7x + 12 = 0 -> need two numbers with product 12, sum 7 -> 3 and 4.
x^2 + 3x + 4x + 12 = 0 -> x(x+3) + 4(x+3) = 0 -> (x+3)(x+4)=0 -> x = -3 or x = -4.
METHOD 2 — QUADRATIC FORMULA (always works)
x = ( -b ± sqrt(b^2 - 4ac) ) / (2a)
Example: 2x^2 - 4x - 3 = 0 (a=2, b=-4, c=-3)
x = ( 4 ± sqrt(16 + 24) ) / 4 = ( 4 ± sqrt(40) ) / 4 = ( 4 ± 6.32 ) / 4 -> x ≈ 2.58 or x ≈ -0.58.
THE DISCRIMINANT (D = b^2 - 4ac) — read the answer before solving:
• D > 0 : two distinct real roots
• D = 0 : one repeated real root
• D < 0 : no real roots (roots are imaginary)
METHOD 3 — COMPLETING THE SQUARE (the idea behind the formula)
x^2 + 6x + 5 = 0 -> x^2 + 6x = -5 -> add (6/2)^2 = 9 to both sides -> (x+3)^2 = 4 -> x+3 = ±2 -> x = -1 or -5.
EXAM TIP: Try factorisation first. If the numbers aren't clean within ~15 seconds, switch to the formula. Always compute D first in word problems — it tells you whether a real solution even exists.