How to Solve Quadratic Equations Step by Step (All 3 Methods)

Every quadratic equation solver — human or app — is running one of three methods: factorisation, completing the square, or the quadratic formula. If you can run all three by hand and know which one to reach for, no quadratic on an exam paper can surprise you. Let's solve the same equation all three ways so the methods can be compared honestly:

x² − 5x + 6 = 0

Method 1: Factorisation (fastest, when it works)

1. Find two numbers that multiply to +6 (the constant) and add to −5 (the x-coefficient).
2. −2 and −3: (−2) × (−3) = 6 ✓ and (−2) + (−3) = −5 ✓
3. Split into brackets: (x − 2)(x − 3) = 0
4. A product is zero when either factor is zero: x = 2 or x = 3.

When to use it:when the numbers are small and the factors jump out within ~20 seconds. If they don't, stop hunting — move to the formula. Students lose exam minutes trying to force a factorisation that doesn't exist (many quadratics simply don't factor neatly).

Method 2: Completing the square

1. Move the constant: x² − 5x = −6
2. Take half the x-coefficient (−5/2 = −2.5), square it (6.25), add to both sides: x² − 5x + 6.25 = 0.25
3. The left side is now a perfect square: (x − 2.5)² = 0.25
4. Square root both sides — both roots: x − 2.5 = ±0.5
5. x = 3 or x = 2. Same answers, as they must be.

When to use it:when the question explicitly demands it, when you need the vertex form for a graph, or when deriving the formula itself. It's rarely the fastest route to roots, but examiners love asking for it directly — so it must be in your hands, not just your notes.

Method 3: The quadratic formula (always works)

For ax² + bx + c = 0:

x = [−b ± √(b² − 4ac)] / 2a

1. Identify a = 1, b = −5, c = 6. Write them down — sign errors here are the #1 mark killer.
2. Discriminant: b² − 4ac = 25 − 24 = 1
3. x = [5 ± √1] / 2 = (5 ± 1)/2
4. x = 3 or x = 2.

When to use it:whenever factorisation doesn't reveal itself quickly, whenever the coefficients are ugly, whenever you're under time pressure. It is never wrong.

The discriminant tells you the story before you solve

• b² − 4ac > 0: two distinct real roots (our example: 1 > 0, two roots).
• b² − 4ac = 0: one repeated real root — the parabola just touches the x-axis.
• b² − 4ac < 0: no real roots (complex roots) — a favourite theory question.

Checking the discriminant first takes ten seconds and prevents the classic disaster of hunting for real factors that don't exist.

Always verify: substitute back

x = 2: (2)² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓. Ten seconds, and it converts “I think I'm right” into “I am right.” This habit — verifying answers instead of trusting the first output — is worth marks in every calculation course you will ever take. (It's also the principle we built into BrainDrill's tutor: it shows each step like the working above and checks its result before presenting it, rather than just sounding confident. If you get stuck on a quadratic at 1am, you can photograph it and walk through the steps together — then re-solve it yourself from a blank page, which is where the learning actually happens.)