Algebra

Quadratic Equation Solver

Enter the three coefficients of a quadratic in standard form. The tool computes the discriminant, substitutes into the quadratic formula, and reports two real roots, one repeated root, or no real roots — whichever is correct.

Inputsax² + bx + c = 0

a (x² term)b (x term)c (constant)

Step-by-step

waiting

Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

2 tools in Algebra.Back to the tool directory →

Worked example

Solve x² − 5x + 6 = 0.

1.
Read off the three coefficients from the equation.
```
x² − 5x + 6 = 0
a = 1, b = −5, c = 6
```
2.
Write down the quadratic formula.
```
x = ( −b ± √(b² − 4ac) ) ÷ 2a
```
3.
Work out the discriminant, the part under the square root. Its sign tells you how many real roots there are.
```
b² − 4ac = (−5)² − 4(1)(6) = 25 − 24 = 1
```
4.
The discriminant is a perfect square, so the square root is a whole number.
```
√1 = 1
```
5.
Substitute everything into the formula, taking the plus and the minus in turn.
```
x = (5 ± 1) ÷ 2
```
6.
Work out each branch separately.
```
x₁ = 6 ÷ 2 = 3
x₂ = 4 ÷ 2 = 2
```

Answer: x = 3 or x = 2

Note: Real roots only. When the discriminant is negative, the tool says there are no real roots.

Use “Load worked example” above to run this in the solver.

Related formulas

The formulas this tool uses, each explained with symbols, conditions and its own example in the Formula Library.

Quadratic formulax = (−b ± √(b² − 4ac)) ÷ (2a)DiscriminantΔ = b² − 4ac