Quadratic Equation Solver

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

How to use Quadratic Equation Solver

1. Enter the coefficients a, b, and c in the respective fields.
2. The roots are calculated instantly using the quadratic formula.
3. Expand 'Show working' to see the step-by-step discriminant and root calculations.
4. Copy the roots in decimal or exact form.
5. Verify the result by substituting each root back into ax² + bx + c — both should yield 0 (within floating-point tolerance).
6. 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

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

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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.