Quadratic Formula Calculator with Steps
The roots of ax² + bx + c = 0 with the discriminant worked out, the factored form and the vertex.
Results
What this tool does
The quadratic formula is the first formula most people meet that has to be memorised rather than derived, and the discriminant under the square root decides everything: positive gives two roots, zero gives one, negative gives a pair of complex ones. This page shows the substitution line by line, so the arithmetic can be checked rather than trusted, and adds what the formula alone does not give you — the factored form, the vertex, the axis of symmetry, and the sum and product of the roots that let you verify the answer in two seconds.
Formula
Δ = b² − 4ac · x = (−b ± √Δ) ÷ 2a · xᵥ = −b ÷ 2a
Variables
| Symbol | Meaning | Unit |
|---|---|---|
a | a | — |
b | b | — |
c | c | — |
SO | The solutions | — |
ST | Step · the substitution · what it gives | — |
KD | What kind of roots | — |
d | Discriminant | — |
x1 | Root x₁ | — |
x2 | Root x₂ | — |
RE | Real part of the complex roots | — |
IM | Imaginary part of the complex roots | — |
FA | Written as a product | — |
VX | The vertex | — |
OP | Which way the parabola opens | — |
SR | The roots added up | — |
PO | The roots multiplied | — |
YI | Y-intercept (b) | — |
Worked example
- a1
- b-3
- c2
- The solutionsx1 = 2, x2 = 1
- Step · the substitution · what it gives D = b^2 - 4ac -3^2 - 4*1*2 1 x = (-b +- sqrt D) / 2a (3 +- sqrt 1) / 2 sqrt D sqrt 1 1 x1 (3 + 1) / 2 2 x2 (3 - 1) / 2 1
- What kind of rootsTwo different real roots
- Discriminant1.00000000
- Root x₁2.0000000000
- Root x₂1.0000000000
- Real part of the complex roots2.0000000000
- Imaginary part of the complex roots0.0000000000
- Written as a product(x - 2)(x - 1)
- The vertex(1.5, -0.25)
- Which way the parabola opensUpwards — the vertex is the lowest point
- The roots added up3.0000000000
- The roots multiplied2.0000000000
- Y-intercept (b)2.00000000
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
Frequently asked questions
What happens when the discriminant is negative?
There is no real solution: the parabola never crosses the horizontal axis. The calculator reports an error rather than a number, because the two roots would be complex. A discriminant of exactly zero means the two roots coincide into a single one.