Quadratic Equation Calculator

Quadratic equations sit at the center of algebra—and the quadratic formula is long enough that one sign mistake ruins the whole pair of roots. I have watched students get the discriminant right and still divide by 2a incorrectly. This page takes a, b, and c in standard form and returns the discriminant, both roots, and a plain-language nature row so you can spot complex answers before copying them into a graphing assignment.

How to Use This Quadratic Equation Calculator

• Write the equation as ax² + bx + c = 0. Move everything to one side so the squared term is positive if possible. Coefficients can be decimals or negatives.
• Enter a, b, and c. Include signs—if the equation is x² − 5x + 6 = 0, then a = 1, b = −5, c = 6.
• Press Calculate. Read discriminant, x₁, x₂, and the nature-of-roots summary in the results table.
• Linear fallback. When a = 0, the tool solves bx + c = 0 instead and labels the case linear. Discriminant shows an em dash because it does not apply.

For factoring practice on simpler polynomials, try the Factor Calculator. For straight exponent and root work, see the Exponent Calculator and Root Calculator.

Quadratic Formulas and Practical Applications

A quadratic graphs as a parabola. The roots are x-intercepts—where the curve hits the axis. Projectile motion, revenue optimization with a square term, and area problems with fixed perimeter all reduce to ax² + bx + c = 0.

Δ = b² − 4ac

x = (−b ± √Δ) / (2a)

Δ > 0: two real crossings. Δ = 0: vertex touches the axis (one repeated root). Δ < 0: no real crossings—complex conjugate roots appear as a ± bi in the table.

Worked real-root example

x² − 5x + 6 = 0 → a = 1, b = −5, c = 6. Δ = 25 − 24 = 1. x = (5 ± 1)/2 → 3 and 2. Factoring would give (x − 3)(x − 2)—the calculator skips to the roots directly.

Complex roots

x² + 1 = 0 → Δ = −4. Roots are ±i. The nature row reads “two complex conjugate roots” so you know not to expect x-intercepts on a real graph.

Linear edge case

2x − 8 = 0 → a = 0, b = 2, c = −8. Single root x = 4. If b = 0 and c ≠ 0 with a = 0, the tool reports no solution—contradiction like 0 = 5.

Coefficients parse from typed text—enter −5 for b rather than relying on a separate sign button. The nature-of-roots row translates the discriminant into plain language so you know whether to expect two x-intercepts, a tangent touch, or complex conjugates before you sketch the parabola.

Real roots print with trimmed decimals; complex roots show real and imaginary parts in standard a ± bi form. This is a numeric solver for standard-form quadratics—not a symbolic factoring engine. When you need to expand (x − 3)(x − 2), the Factor Calculator handles the opposite direction.

Projectile problems often produce quadratics in time: height = −16t² + v₀t + h₀ in US customary units. Enter the coefficients as a, b, c after rearranging to standard form equals zero. Positive time roots are the physically meaningful ones; negative time roots still appear in the table but may be discarded in context.

When Δ is a perfect square, roots may still be decimals if a ≠ 1—always trust the table over mental factoring. Complex roots display with imaginary unit i; add them only if your course accepts non-real solutions in the final answer box.

Frequently Asked Questions

Scan the nature-of-roots row before you copy x₁ and x₂ into a graphing tool. Repeated roots mean the vertex touches the axis; complex roots mean no real intercepts. When a = 0, treat the output as a linear solver—discriminant shows an em dash because b² − 4ac no longer applies.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / (2a) for a ≠ 0. It always works when factoring is messy—enter coefficients and let the table compute.

What happens when a equals zero?

You get a linear equation. One root x = −c/b when b ≠ 0; identity or no solution when b = 0 depending on c.

What does a negative discriminant mean?

No real roots—the parabola floats above or below the axis without crossing. Complex conjugate roots still satisfy the equation algebraically.

Can coefficients be decimals or fractions?

Yes—type them as text. High-precision parsing avoids premature rounding before Δ and the square root are evaluated.

Why do x₁ and x₂ match when the discriminant is zero?

Repeated root at the vertex. Both rows show −b/(2a)—the parabola touches the axis once instead of crossing twice.

Disclaimer. RapidRatio is informational only—not exam authority or computer-algebra replacement. Show work as your instructor requires.