There's more than one way to crack a quadratic, and a good quadratic equation solver lets you check whichever you choose. We'll solve x² − 5x + 6 = 0 four different ways and get x = 2, 3 every time.
On this page
Method 1: Factoring
Find two numbers that multiply to c and add to b. For x² − 5x + 6, that's −2 and −3:
x² − 5x + 6 = (x − 2)(x − 3) = 0 → x = 2 or x = 3Fast when the roots are tidy integers, useless when they aren't.
Method 2: Completing the square
Rewrite as a perfect square plus a constant:
x² − 5x + 6 = (x − 2.5)² − 0.25 = 0
(x − 2.5)² = 0.25 → x − 2.5 = ±0.5 → x = 3 or 2This method also hands you the vertex (2.5, −0.25) for free, which is why it underpins the formula itself.
Method 3: The quadratic formula
x = ( −b ± √(b² − 4ac) ) / (2a)With a = 1, b = −5, c = 6: x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2 = 3 or 2. The formula never fails — and when b² − 4ac is negative it cleanly produces complex roots.
2x² + 3x − 5 = 0 → x = (−3 ± √49)/4 = (−3 ± 7)/4 = 1 or −2.5.
Method 4: The EQN solver
Switch to EQN mode, choose a degree-2 polynomial, and enter the coefficients a, b, c:
1, −5, 6 → x = 2, x = 3Instant, and it returns complex roots when they occur. See the equation solver pillar guide for cubic and simultaneous equations too.
Which method when
| Method | Best when |
|---|---|
| Factoring | Roots are simple integers |
| Completing the square | You also need the vertex / max-min |
| Quadratic formula | Anything — especially messy or complex roots |
| EQN solver | Checking your answer instantly |
A quick b² − 4ac tells you whether to expect two real, one repeated, or complex roots before you commit to a method.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) / (2a) — it solves any quadratic, including complex-root cases.
Which method should I use?
Factor for tidy roots, complete the square for the vertex, use the formula otherwise — and check with EQN.
How do I know if roots are real?
The discriminant b²−4ac: positive → two real, zero → one repeated, negative → complex pair.
Check your quadratic
Enter a, b, c in EQN mode and confirm your roots in one step.
Open the quadratic equation solver →