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ToolBoxGenie

Quadratic Equation Solver

Calculators · Added 15 August 2026

Enter the three coefficients of a quadratic and this solver returns the roots by the quadratic formula, along with the discriminant that determines their nature, the vertex of the parabola, its axis of symmetry, and a factored form where the roots are tidy. Complex roots are reported properly rather than dismissed as no solution.

For an equation in the form ax² + bx + c = 0.

Cannot be zero.

Blank counts as zero.

Blank counts as zero.

How to use the quadratic equation solver

  1. 1Write your equation in the form ax² + bx + c = 0, moving everything to one side first if needed.
  2. 2Enter a, the coefficient of x². It cannot be zero — that would make the equation linear.
  3. 3Enter b and c. Leave either blank to treat it as zero.
  4. 4Press Solve.
  5. 5Read the discriminant first: it tells you what kind of roots to expect before you look at them.

Examples

Two real roots

Input
x² − 5x + 6 = 0
Result
x = 3 and x = 2 · discriminant 1 · factors as (x − 3)(x − 2)

A positive discriminant means the parabola crosses the x-axis twice.

One repeated root

Input
x² + 2x + 1 = 0
Result
x = −1 · discriminant 0

The parabola touches the axis without crossing. The vertex sits exactly on the root.

Complex roots

Input
x² + 1 = 0
Result
x = 0 + 1i and x = 0 − 1i · discriminant −4

No real solution, but the roots exist. The parabola never reaches the x-axis.

About the quadratic equation solver

Where the quadratic formula comes from

The formula x = (−b ± √(b² − 4ac)) ÷ 2a is not arbitrary — it is what falls out of completing the square on the general equation, done once so nobody has to do it again for each specific case.

The process: divide through by a to make the leading coefficient 1, move the constant across, add the square of half the x-coefficient to both sides to make the left side a perfect square, then take the square root. Every part of the final formula traces back to a step in that sequence. The −b/2a is the halving; the ± is the two square roots; the discriminant is whatever is left under the radical.

That derivation also explains why the vertex sits at −b/2a. Completing the square rewrites the quadratic as a(x − h)² + k, and h is the value that makes the squared term vanish — the same −b/2a. The formula and the vertex are two views of the same manipulation.

Checking an answer without redoing the work

Two relationships between the roots and the coefficients make verification quick, and both are shown in the results. For any quadratic, the sum of the roots equals −b/a and their product equals c/a. These follow directly from expanding a(x − r₁)(x − r₂) and comparing coefficients.

They are genuinely useful as a check. For x² − 5x + 6, the roots 3 and 2 sum to 5, which matches −(−5)/1, and multiply to 6, which matches 6/1. A mistyped coefficient almost always breaks one of these two identities, which makes them faster than re-running the whole formula.

They also make some problems solvable by inspection. Asked for two numbers that sum to 7 and multiply to 12, you have factored x² − 7x + 12 without touching the formula. This is what people are doing when they factor quadratics 'by looking at them', and it works well precisely when the roots are small integers — which is exactly when the calculator offers a factored form too.

Frequently asked questions

What is the discriminant and why does it matter?
It is b² − 4ac, the part under the square root in the quadratic formula, and it decides the nature of the roots before you calculate them. Positive gives two distinct real roots, zero gives one repeated root, negative gives a complex conjugate pair. Checking it first tells you what kind of answer to expect and catches sign errors early.
Why does it say complex instead of no solution?
Because 'no solution' is only true if you restrict yourself to real numbers. Every quadratic has exactly two roots counted with multiplicity — that is the fundamental theorem of algebra — and when the discriminant is negative those roots involve the imaginary unit i. They are perfectly real solutions in the complex plane, and they matter enormously in engineering and physics.
What is the vertex and how is it found?
The turning point of the parabola, at x = −b ÷ 2a. It is a minimum when a is positive and a maximum when a is negative, which makes it the answer to most optimisation problems that reduce to a quadratic — maximum area, minimum cost, peak height of a projectile. The axis of symmetry is the vertical line through it.
Why is there no factored form sometimes?
Because it would be less readable than the roots. A factored form is only offered when the roots are clean rationals; when they are long decimals, writing (x − 2.7320508…)(x + 0.7320508…) tells you nothing the root list did not. Complex roots are not factored over the reals at all.
What if a is zero?
Then it is not a quadratic. With a = 0 the equation becomes bx + c = 0, a linear equation with the single solution x = −c/b. The calculator rejects it rather than dividing by zero, and says why.

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