MathBench

Quadratic Equation Solver

Enter the three coefficients and get the roots — in simplified exact form and as decimals — plus the discriminant, vertex, and factored form, with the solution worked step by step.

x² − 3x − 10 = 0

Solution

—root x₁
—root x₂
—discriminant b² − 4ac
—vertex (h, k)

How this calculator works

Type the three coefficients of ax² + bx + c = 0 and the solver applies the quadratic formula with its work shown: the discriminant first (which announces what kind of roots to expect), then the substitution, the simplified radical when the discriminant isn't a perfect square, and finally the roots in both exact and decimal form. Rational roots trigger the factored form; a negative discriminant produces the complex conjugate pair rather than an error.

The vertex comes along for free — h = −b/2a, k from substituting back — because most homework that starts with the roots ends with a sketch. If your assignment says "solve by completing the square," use the companion calculator, which walks that method on the same equation.

The formula

x = ( −b ± √(b² − 4ac) ) / 2a

Δ = b² − 4ac      Δ > 0: two real roots
                  Δ = 0: one repeated root
                  Δ < 0: complex conjugate pair
Vertex: h = −b/2a,  k = c − b²/4a

The formula is completing the square performed once on the general equation — the derivation is shown in full in Paul's Online Math Notes (Lamar University), Algebra: Quadratic Equations Part II, a freely available university reference. The radical simplification follows the standard convention of extracting the largest perfect square factor and reducing the common factor of the fraction.

Worked example

Solve x² − 3x − 10 = 0 (the calculator's default):

  1. Identify: a = 1, b = −3, c = −10
  2. Discriminant: (−3)² − 4(1)(−10) = 9 + 40 = 49
  3. √49 = 7 — a perfect square, so the roots are rational
  4. x = (3 ± 7) / 2 → x₁ = 5, x₂ = −2
  5. Factored: (x − 5)(x + 2) = 0 — expand it to check
  6. Vertex: h = 3/2, k = −10 − 9/4 = (1.5, −12.25)

Try changing c to −1: the discriminant becomes 13, not a perfect square, and the solver reports the exact roots (3 ± √13)/2 with their decimals — the form a textbook answer key prints.

Assumptions & tips

  • Read the discriminant before the roots. It's the fastest error check there is: if your hand calculation produced two real roots but Δ is negative, the mistake is upstream of the formula.
  • Keep signs in parentheses. The classic error is squaring −3 as −9. Write (−3)² and the parentheses do the remembering for you — the calculator's steps display does exactly this.
  • Check by substitution or by Vieta. The roots must satisfy x₁ + x₂ = −b/a and x₁ · x₂ = c/a. For the default: 5 + (−2) = 3 ✓ and 5 × (−2) = −10 ✓ — a ten-second verification.
  • Exact form matters in class, decimals in the shop.(3 + √13)/2 is the answer for homework; 3.302776 is the answer for cutting material. This solver gives both so you never trade one for the other.
  • Quadratics hide in word problems. Projectile height, area optimizations, and break-even revenue all reduce to ax² + bx + c = 0. Set the expression equal to zero first — the formula only solves equations, not expressions.

Frequently asked questions

What does the discriminant tell you?

Everything about the character of the roots before you compute them. The discriminant is b² − 4ac: positive means two distinct real roots, zero means one repeated real root (the parabola just touches the x-axis), and negative means a pair of complex conjugate roots (the parabola never crosses the axis). If it is a perfect square and the coefficients are integers, the roots are rational and the quadratic factors cleanly.

Where does the quadratic formula come from?

From completing the square on the general equation ax² + bx + c = 0 — divide by a, move the constant, add the square of half the x-coefficient to both sides, and solve. The formula is that derivation done once, in general, so nobody has to repeat it per problem. Our complete-the-square calculator walks the same process step by step on your specific numbers.

What are complex roots, and are they real answers?

When the discriminant is negative, its square root involves i = √(−1), and the two roots form a conjugate pair like −1 + 2i and −1 − 2i. They are perfectly legitimate solutions — the parabola simply never touches the x-axis, so the solutions live off the real number line. In physics and engineering, complex roots of characteristic equations describe oscillation, so they carry real meaning.

Why does the calculator show exact forms like (3 + √17)/4?

Because a decimal like 1.780776 is an approximation, and mathematics teachers (rightly) want the exact value. When the discriminant is not a perfect square, the calculator simplifies the radical — pulling out square factors — and reduces the resulting fraction, giving the same simplified exact answer a textbook expects, alongside the decimal for practical use.

What if a = 0?

Then the equation is not quadratic at all — it collapses to the linear equation bx + c = 0 with the single solution x = −c/b. The calculator handles this case and says so explicitly, because silently applying the quadratic formula would divide by zero.

Sources

  1. NIST Digital Library of Mathematical Functions — §1.11 Zeros of Polynomials — National Institute of Standards and Technology (DLMF release of 2026-06-15). dlmf.nist.govGives the roots of az² + bz + c = 0 as (−b ± √D)/2a with D = b² − 4ac, and the sum and product of the roots (−b/a and c/a) used as the Vieta check in the tips.
  2. Algebra - Quadratic Equations - Part II — Paul's Online Notes, Paul Dawkins, Lamar University, 2003–2026. tutorial.math.lamar.eduThe derivation of the quadratic formula by completing the square that the formula section above cites.
  3. Algebra - Quadratic Equations : A Summary — Paul's Online Notes, Paul Dawkins, Lamar University, 2003–2026. tutorial.math.lamar.eduDefines the discriminant and its three cases — two real roots, one repeated root, a complex conjugate pair — which is exactly how this solver classifies its answers.