Two real roots: x² - 5x + 6 = 0
Input: a = 1, b = -5, c = 6
Output: Δ = 1; x = 3 or x = 2 (factored form: (x-3)(x-2) = 0)
UtilityKit
500+ fast, free tools. Most run in your browser only; Image & PDF tools upload files to the backend when you run them.
Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 using the quadratic formula, showing real and complex roots with full working.
About Quadratic Equation Solver
The Quadratic Equation Solver finds the roots of any quadratic equation in the form ax² + bx + c = 0 using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The tool calculates the discriminant (b² - 4ac) and determines whether the equation has two distinct real roots (discriminant > 0), one repeated real root (discriminant = 0), or two complex conjugate roots (discriminant < 0). All three cases are handled and explained. The step-by-step working shows the discriminant calculation, the ± solution substitution, and the simplified final roots — making this ideal for algebra students learning the quadratic formula. The quadratic formula is one of the oldest closed-form algebraic results, attributable to Babylonian and Indian mathematicians millennia before its modern symbolic form.
Why use Quadratic Equation Solver
- Handles all three discriminant cases: two real roots, one root, and complex roots.
- Step-by-step working helps students follow the quadratic formula.
- Shows both exact (radical) form and decimal approximations.
- Instant calculation — no form submission needed.
- Detailed working makes it ideal for algebra homework where step-by-step solutions are required, not just final answers.
- Privacy-first: coefficients are processed entirely in your browser.
How to use Quadratic Equation Solver
- Enter the coefficients a, b, and c in the respective fields.
- The roots are calculated instantly using the quadratic formula.
- Expand 'Show working' to see the step-by-step discriminant and root calculations.
- Copy the roots in decimal or exact form.
- Verify the result by substituting each root back into ax² + bx + c — both should yield 0 (within floating-point tolerance).
- Use the discriminant first if you only need to know whether real roots exist — full computation is unnecessary for that question.
When to use Quadratic Equation Solver
- Solving quadratic equations in algebra coursework.
- Checking hand-calculated roots for errors.
- Finding roots of physics, engineering, or economics quadratic models.
- Learning the quadratic formula with step-by-step examples.
- Solving projectile motion or trajectory problems in physics (x represents time or distance).
- Computing break-even points in business and economics where revenue equals cost.
Examples
Repeated root: x² - 4x + 4 = 0
Input: a = 1, b = -4, c = 4
Output: Δ = 0; x = 2 (repeated). Factored: (x-2)² = 0
Complex roots: x² + x + 1 = 0
Input: a = 1, b = 1, c = 1
Output: Δ = -3; x = -0.5 ± 0.866i
Tips
- Always check whether your equation is in standard form (ax² + bx + c = 0) before entering coefficients — moving terms across the equals sign changes signs.
- If the discriminant is a perfect square, the roots are rational and can also be found by factoring — try factoring first for cleaner answers.
- For physics problems (projectile motion, kinematics), the negative root is often physically meaningless (negative time) — pick the positive one.
- When |b| is much larger than |c|, beware floating-point cancellation; the alternative formula x = -2c / (b + sign(b)√(b²-4ac)) gives more accurate small roots.
- The sum of the roots equals -b/a and the product equals c/a — use these as a sanity check after computing.
- Complex roots always come in conjugate pairs (a + bi and a - bi) for real-coefficient quadratics — if one root is complex, the other is its conjugate.
Frequently Asked Questions
What is the quadratic formula?▾
x = (-b ± √(b² - 4ac)) / 2a, where a, b, c are the coefficients of ax² + bx + c = 0.
What is the discriminant?▾
The discriminant is b² - 4ac. If > 0: two distinct real roots. If = 0: one repeated real root. If < 0: two complex conjugate roots.
Can a be zero?▾
No. If a = 0, the equation is linear (bx + c = 0), not quadratic. The solver requires a ≠ 0.
What are complex roots?▾
When the discriminant is negative, the roots include √(negative number) = i × √(|discriminant|/4a²). The roots are a ± bi where a and b are real numbers.
How do I convert the answer to factored form?▾
If roots are x₁ and x₂, factored form is a(x − x₁)(x − x₂) = 0. This is shown in the working output.
How does this compare to a graphing calculator's solver?▾
Identical math, but this tool shows the discriminant, vertex, factored form, and step-by-step working in plain text — useful for homework explanations. A TI-84 just gives numerical roots.
How accurate are the decimal approximations?▾
JavaScript double-precision floats give ~15-17 significant digits. For extremely small/large coefficients or near-zero discriminants, precision can degrade — the tool flags potentially unstable cases.
Does this handle non-real coefficients?▾
No — coefficients must be real. Complex coefficients require fundamentally different handling (the discriminant itself becomes complex). For real-coefficient cases with complex roots, this tool works correctly.
Explore the category
Glossary
- Quadratic equation
- An equation of the form ax² + bx + c = 0 with a ≠ 0. Has at most two solutions (roots).
- Discriminant
- Δ = b² - 4ac. The expression under the square root in the quadratic formula. Sign determines the nature of the roots.
- Quadratic formula
- x = (-b ± √(b² - 4ac)) / 2a. Gives both roots of any quadratic; provable by completing the square.
- Roots / zeros
- Values of x where the quadratic equals zero. Geometrically, where the parabola crosses the x-axis.
- Vertex
- The maximum or minimum point of the parabola; located at x = -b/2a, y = c - b²/4a.
- Complex conjugate
- If a + bi is a root of a real-coefficient quadratic, so is a - bi. They mirror across the real axis in the complex plane.
- Completing the square
- Algebraic technique that rewrites ax² + bx + c as a(x + b/2a)² + (c - b²/4a) — the geometric basis of the quadratic formula.