A quadratic equation is any equation where the highest power of x is 2 — like x² − 5x + 6 = 0. There are exactly three reliable methods to solve them. Knowing all three — and when to use each — means you will never be stuck.
Standard Form
When you solve a quadratic, you are finding the roots — the values of x where the parabola crosses the x-axis. A quadratic can have two roots, one root (a repeated root), or no real roots at all.
Not sure which quadratic method to use first?
A free demo session lets us pinpoint exactly where the reasoning breaks down — no pressure, no commitment.
Book free demoMethod 1 — Factoring
Factoring works best when the equation has nice integer roots. You rewriteax² + bx + c as a product of two brackets, then use the zero-product property.
Solve by factoring: x² − 5x + 6 = 0
Find two numbers that multiply to +6 and add to −5
Answer: −2 and −3 (since −2 × −3 = 6 and −2 + −3 = −5).
Write the factored form
Apply zero-product property
Check both solutions
x = 3: 9 − 15 + 6 = 0 ✓
Solve by factoring: 2x² + 7x + 3 = 0
Multiply a × c = 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7
Split the middle term and factor by grouping
Solve each factor
x + 3 = 0 → x = −3
Method 2 — The Quadratic Formula
When factoring is difficult or impossible, the quadratic formula always works. Memorise this formula — it solves any quadratic.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
where ax² + bx + c = 0
Solve using the formula: 2x² − 4x − 6 = 0
Identify a, b, and c
Calculate the discriminant (b² − 4ac)
Apply the formula
Calculate both solutions
x = (4 − 8)/4 = −4/4 = −1
The Discriminant (b² − 4ac)
- • b² − 4ac > 0 → two different real roots
- • b² − 4ac = 0 → exactly one real root (repeated)
- • b² − 4ac < 0 → no real roots
Method 3 — Completing the Square
Completing the square is more steps but gives you the vertex form of the parabola, which is useful in more advanced work. It is also how the quadratic formula was derived.
Solve by completing the square: x² + 6x − 7 = 0
Move the constant to the right side
Take half of the x-coefficient, square it, add to both sides
Write the left side as a perfect square
Take the square root of both sides (±)
Which Method Should You Use?
| Situation | Best method |
|---|---|
| Looks like it will factor cleanly | Factoring (fastest) |
| Messy coefficients or fractions | Quadratic formula (always works) |
| Need vertex form for graphing | Completing the square |
| Not sure | Quadratic formula (safest default) |
Practice Problems
- 1
Solve by factoring: x² − 7x + 12 = 0
Hint: Find two numbers that multiply to 12 and add to −7.
- 2
Solve using the quadratic formula: 3x² − 5x − 2 = 0
Hint: a=3, b=−5, c=−2.
- 3
Solve by completing the square: x² + 4x − 5 = 0
Hint: Add (4/2)² = 4 to both sides.
- 4
Find the discriminant and state the number of roots: 2x² − 3x + 5 = 0
Hint: b² − 4ac = 9 − 40 = ?
- 5
Solve: x² − 9 = 0
Hint: This is a difference of squares — factor it as (x+3)(x−3) = 0.
Want to practise with feedback?
Quadratics are one of those topics where small mistakes compound into bigger ones — especially under exam pressure. A free demo session lets us find exactly which step is tripping you up and fix it before it becomes a habit.
