Quadratic equation solver

Solve any quadratic equation ax² + bx + c = 0 in seconds

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. This solver computes the discriminant Δ, tells you how many solutions exist, gives the roots as decimals and in exact form (simplified radicals, reduced fractions, complex numbers), and draws the parabola with its roots and vertex.

Quadratic equations are a core topic in Algebra 1 and Algebra 2, GCSE and A-level maths: factoring, completing the square and the quadratic formula all lead to the same roots. Students use the tool to check homework, teachers to build answer keys, and physics, engineering or economics students to solve equations that come from real problems such as projectile motion, break-even points or areas.

1. Enter a, b and c

   Type the coefficients into “a (coefficient of x²)”, “b (coefficient of x)” and “c (constant term)”. Decimals and negative numbers are accepted, and the “Your equation” line shows the equation exactly as it was read.

2. Read Δ and the solutions

   The purple box shows the discriminant, what its sign means (two roots, a double root or complex roots), x₁ and x₂ rounded to 6 decimal places and the “Exact form”. The “Copy result” button puts everything on your clipboard.

3. Use the cards and the graph

   The cards show the vertex (h, k), axis of symmetry, y-intercept, opening, sum and product of the roots, vertex form and factored form; the graph marks the roots, the vertex and the y-intercept.

4. Open the step-by-step solution

   Expand “Step-by-step solution” to see Δ computed with your numbers, the quadratic formula with values substituted and the vertex calculation, ready to copy into your work.

The method: discriminant and quadratic formula

For ax² + bx + c = 0 (a ≠ 0), first compute Δ = b² − 4ac. If Δ > 0 the roots are x₁ = (−b − √Δ)/(2a) and x₂ = (−b + √Δ)/(2a); if Δ = 0 the double root is x = −b/(2a); if Δ < 0 there is no real root but two complex conjugate ones.

Completing the square gives the vertex form a(x − h)² + k with h = −b/(2a) and k = −Δ/(4a): the parabola is the graph of y = x² stretched vertically by a factor a (and flipped if a < 0), then shifted so that its vertex sits at (h, k). Under the hood, the sign of Δ is tested with exact integer arithmetic (so 0.1x² + 0.2x + 0.1 = 0 is correctly seen as a double root) and the roots use the numerically stable form of the formula, which avoids losing digits when b² is much larger than 4ac.

• a, b, c: the coefficients; a multiplies x² and must be non-zero, b multiplies x, c is the constant term.
• Δ (discriminant): its sign gives the number of real roots (2, 1 or 0); when Δ < 0 the roots are p ± qi with p = −b/(2a) and q = √(−Δ)/(2|a|).
• ±: the minus sign gives one root, the plus sign the other; the whole numerator −b ± √Δ is divided by 2a.
• Vertex (h, k): the minimum of the function if a > 0, the maximum if a < 0; the line x = h is the axis of symmetry.
• Sum and product: x₁ + x₂ = −b/a and x₁ × x₂ = c/a (Vieta’s formulas), handy to check a result or spot an obvious root.
• Exact form: √Δ is simplified by taking square factors out of the radical (√48 = 4√3), then the fraction is reduced by the GCD of its three terms.

Worked examples

What is a quadratic equation solver used for?

• Homework and exams: check Δ, the roots and the vertex form of an exercise, step by step.
• Physics: a ball thrown straight up at 20 m/s from a height of 1.5 m follows h(t) = −4.9t² + 20t + 1.5. Solving h(t) = 0 gives t ≈ 4.155303 s (the other, negative root is rejected), and the vertex gives a maximum height of about 21.908163 m at t ≈ 2.040816 s.
• Geometry: a rectangle with perimeter 20 and area 21 has sides that are the roots of x² − 10x + 21 = 0, namely 3 and 7.
• Business: a profit P(q) = −2q² + 40q − 150 is zero at q = 5 and q = 15 and peaks at 50 for q = 10, read straight from the vertex.
• Teaching: build examples with integer roots and show in class how a, b and c reshape the parabola.

Common mistakes

• Dropping the sign of b: in x² − 3x + 2, b = −3, so −b = 3 and b² = (−3)² = 9, not −9.
• Dividing only part of the numerator by 2a: the whole −b ± √Δ is divided, not just √Δ.
• Reading a, b, c before rearranging: 2x² = 7x − 3 must first become 2x² − 7x + 3 = 0.
• Saying “no solution” when Δ < 0: there is no real solution, but there are two complex conjugate ones.

Everything is computed locally in your browser: no data is sent anywhere. Decimal results are rounded to 6 decimal places; the exact form is shown as long as the absolute value of the discriminant, computed with the coefficients scaled to integers, stays below 2⁵³ ≈ 9 × 10¹⁵. Non-zero coefficients must lie between 10⁻¹⁰⁰ and 10¹⁰⁰ in absolute value.