Algebra & Functions

Quadratic Equation Solver

Enter the coefficients a, b and c of ax² + bx + c = 0. The solver computes the discriminant, applies the quadratic formula, and checks each root by substituting it back.

Updated October 3, 2026Written and reviewed by Mateuss M.Method: how we calculate

Try an example:

Answerx = 4 or x = −1

Factored form: (x − 4)(x + 1)
Discriminant D = 25
Vertex: (1.5, −6.25)

Step-by-step working 6 steps

Identify a, b and c

x² − 3x − 4 = 0
a = 1,  b = −3,  c = −4

2
Calculate the discriminant
```
D = b² − 4ac = (−3)² − 4(1)(−4) = 25
```
D is positive: two different real roots.

Apply the quadratic formula

x = (−b ± √D) / 2a = (3 ± √25) / 2 = (3 ± 5) / 2

4
Solve for both roots
```
x₁ = 4
x₂ = −1
```
5
Check by substitution
```
f(4) = 0
f(−1) = 0
```
Both should be 0 (allowing for tiny rounding).
6
Vertex of the parabola
```
x = −b / 2a = 1.5,  y = −6.25  →  vertex (1.5, −6.25)
```
The parabola opens upward, so the vertex is the minimum point.

How to use the quadratic equation solver

Rearrange your equation so one side is 0 and read off a, b and c.
Enter them. Include signs: −3, not 3, if the term is subtracted.
Read the roots; open the steps for the discriminant and the check.
If the roots are whole numbers, the factored form is shown too.

Quadratic formula

x = (−b ± √(b² − 4ac)) / 2a
Discriminant D = b² − 4ac
Vertex: x = −b / 2a

D > 0: two real roots · D = 0: one repeated root · D < 0: two complex roots.

Worked examples

Each example below is generated by the same calculator you used above, so the working always matches the answer.

x² − 3x − 4 = 0

Answerx = 4 or x = −1

Factored form: (x − 4)(x + 1)
Discriminant D = 25
Vertex: (1.5, −6.25)

Step-by-step working 6 steps

Identify a, b and c

x² − 3x − 4 = 0
a = 1,  b = −3,  c = −4

2
Calculate the discriminant
```
D = b² − 4ac = (−3)² − 4(1)(−4) = 25
```
D is positive: two different real roots.

Apply the quadratic formula

x = (−b ± √D) / 2a = (3 ± √25) / 2 = (3 ± 5) / 2

4
Solve for both roots
```
x₁ = 4
x₂ = −1
```
5
Check by substitution
```
f(4) = 0
f(−1) = 0
```
Both should be 0 (allowing for tiny rounding).
6
Vertex of the parabola
```
x = −b / 2a = 1.5,  y = −6.25  →  vertex (1.5, −6.25)
```
The parabola opens upward, so the vertex is the minimum point.

2x² + 3x − 2 = 0

Answerx = 0.5 or x = −2

Factored form: 2(x − 0.5)(x + 2)
Discriminant D = 25
Vertex: (−0.75, −3.125)

Step-by-step working 6 steps

Identify a, b and c

2x² + 3x − 2 = 0
a = 2,  b = 3,  c = −2

2
Calculate the discriminant
```
D = b² − 4ac = (3)² − 4(2)(−2) = 25
```
D is positive: two different real roots.

Apply the quadratic formula

x = (−b ± √D) / 2a = (−3 ± √25) / 4 = (−3 ± 5) / 4

4
Solve for both roots
```
x₁ = 0.5
x₂ = −2
```
5
Check by substitution
```
f(0.5) = 0
f(−2) = 0
```
Both should be 0 (allowing for tiny rounding).
6
Vertex of the parabola
```
x = −b / 2a = −0.75,  y = −3.125  →  vertex (−0.75, −3.125)
```
The parabola opens upward, so the vertex is the minimum point.

x² + 2x + 1 = 0

Answerx = −1 (double root)

Discriminant D = 0
Vertex: (−1, 0)

Step-by-step working 4 steps

1
Identify a, b and c
```
x² + 2x + 1 = 0
a = 1,  b = 2,  c = 1
```
2
Calculate the discriminant
```
D = b² − 4ac = (2)² − 4(1)(1) = 0
```
D is zero: one repeated real root.
3
One repeated root
```
x = −b / 2a = −2 / 2 = −1
```
4
Vertex of the parabola
```
x = −b / 2a = −1,  y = 0  →  vertex (−1, 0)
```
The parabola opens upward, so the vertex is the minimum point.

x² + 2x + 5 = 0

Answerx = −1 ± 2i

No real x-intercepts: the parabola never crosses the x-axis.
Discriminant D = −16
Vertex: (−1, 4)

Step-by-step working 4 steps

1
Identify a, b and c
```
x² + 2x + 5 = 0
a = 1,  b = 2,  c = 5
```
2
Calculate the discriminant
```
D = b² − 4ac = (2)² − 4(1)(5) = −16
```
D is negative: no real roots — two complex roots.
3
Complex roots
```
x = (−2 ± √16·i) / 2
x = −1 ± 2i
```
The square root of a negative number introduces i, where i² = −1.
4
Vertex of the parabola
```
x = −b / 2a = −1,  y = 4  →  vertex (−1, 4)
```
The parabola opens upward, so the vertex is the minimum point.

Common mistakes to avoid

Dropping a sign on b or c

In x² − 3x − 4 = 0 the coefficients are b = −3 and c = −4. Most wrong answers start here.

Not setting the equation to zero first

x² = 3x + 4 must become x² − 3x − 4 = 0 before reading a, b and c.

Dividing only part of the formula by 2a

The whole numerator, −b ± √D, is divided by 2a — not just the square root.

Frequently asked questions

What if a is 0?

Then it is not quadratic. The solver points you to the linear equation solver.

What does a negative discriminant mean?

There are no real solutions. The parabola never touches the x-axis and the roots are complex numbers.

Should I factor or use the formula?

Factor when the numbers are small and obvious. The formula always works, which makes it the safe default.