CALTRX

Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Shows discriminant, real and complex roots with full working.

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Enter your values above to see the results.

Tips & Notes

  • ✓Check the discriminant first. If b²−4ac < 0, no real solution exists — stop before computing.
  • ✓When Δ is a perfect square (1, 4, 9, 25...) the roots are rational and the equation factors cleanly.
  • ✓Sign errors with b are the most common mistake. For 2x²−6x+4=0, b=−6 so −b=+6.
  • ✓The axis of symmetry x=−b/2a is always the midpoint between the two roots.
  • ✓Use Vieta: roots must sum to −b/a and multiply to c/a. Fast way to check your answer.

Common Mistakes

  • ✗Entering b without its sign. For x²−5x+6=0, b=−5 not 5. Wrong sign gives completely wrong roots.
  • ✗Computing only one root — the ± means always evaluate both (−b+√Δ)/2a and (−b−√Δ)/2a.
  • ✗Dividing only √Δ by 2a instead of the full numerator (−b ± √Δ) / 2a.
  • ✗Rounding the discriminant before taking the square root — keep full precision until the final step.
  • ✗Assuming Δ < 0 means no solution — it means no REAL solution. Complex roots still exist.

Quadratic Formula Calculator Overview

The quadratic formula is the universal solution to any equation of the form ax² + bx + c = 0. Unlike factoring (which only works when roots happen to be rational) or completing the square (which requires careful algebraic manipulation), the quadratic formula always works — it gives exact answers for any real coefficients, whether the roots are integers, fractions, irrational surds, or complex numbers. Memorizing and correctly applying this formula is one of the highest-value skills in algebra.

x = (−b ± √(b² − 4ac)) / 2a
EX: x² − 5x + 6 = 0 → a=1, b=−5, c=6 → x = (5 ± √(25−24))/2 = (5±1)/2 → x=3 or x=2
EX: 2x² + 3x − 2 = 0 → a=2, b=3, c=−2 → x = (−3 ± √(9+16))/4 = (−3±5)/4 → x=1/2 or x=−2
The discriminant Δ = b² − 4ac — check this first to know what kind of solution you'll get:
EX: 3x²−2x+5=0 → Δ=4−60=−56 < 0 → no real roots (complex: x = (2±√(−56))/6 = 1/3 ± i√56/6)
Discriminant interpretation: - Δ > 0 and perfect square → two rational roots, equation factors over integers - Δ > 0, not perfect square → two irrational roots involving √ - Δ = 0 → one repeated root x = −b/2a (the vertex touches the x-axis) - Δ < 0 → two complex conjugate roots (no x-intercepts) Vieta's formulas — use these to verify roots without substituting back:
EX: For x²−7x+12=0, roots should sum to 7 and multiply to 12. Solving: x=3 or x=4. Sum=7✓ Product=12✓
The discriminant Δ = b² − 4ac fully determines the nature of the solutions before solving. Positive discriminant: two distinct real roots. Zero discriminant: one repeated real root — the parabola is tangent to the x-axis at exactly one point. Negative discriminant: no real roots — the parabola does not cross the x-axis, and the solutions are complex conjugates. Checking the discriminant first avoids computing a square root of a negative number unexpectedly. The vertex of the parabola y = ax² + bx + c lies at x = −b/2a — directly from the formula before the ± branch. The two roots are symmetric about this vertex, each located a distance of √Δ/|2a| from it. This geometric interpretation explains why repeated roots (Δ = 0) produce a single point of tangency rather than two crossing points, and why complex roots (Δ < 0) correspond to a parabola that never reaches the x-axis.

Frequently Asked Questions

The discriminant is b² − 4ac. If positive: two distinct real roots (parabola crosses x-axis twice). If zero: one repeated real root (parabola touches x-axis at exactly one point). If negative: two complex conjugate roots (parabola does not cross the x-axis at all). Example: x² − 5x + 6 → discriminant = 25 − 24 = 1 (positive, two real roots: x = 2 and x = 3). x² − 2x + 1 → discriminant = 4 − 4 = 0 (one root: x = 1).

Factoring is faster when the equation factors cleanly with small integers. x² − 5x + 6 = (x−2)(x−3) → roots 2 and 3 — found by inspection. Use the quadratic formula when the equation does not factor easily, when coefficients are large or decimal, or when you need exact irrational values like (3 + √5)/2. The formula works for every quadratic — factoring only works when the roots happen to be rational. When in doubt, use the formula.

The sum of roots = −b/a and the product of roots = c/a (Vieta's formulas). Example: 2x² + 3x − 5 = 0. Sum of roots = −3/2 = −1.5. Product of roots = −5/2 = −2.5. Roots are 1 and −2.5: sum = 1 + (−2.5) = −1.5 ✓, product = 1 × (−2.5) = −2.5 ✓. These relationships let you verify computed roots without substituting back into the original equation.

Complex roots occur when the discriminant b² − 4ac is negative. The roots are x = (−b ± i√|b²−4ac|) / (2a), where i = √−1. Complex roots always come in conjugate pairs: if a + bi is a root, then a − bi is also a root. Example: x² + x + 1 = 0 → discriminant = 1 − 4 = −3 → roots = (−1 ± i√3)/2. These roots appear naturally in electrical circuit analysis, signal processing, and control systems.

Divide every term by the leading coefficient to make it x² + (b/a)x + (c/a) = 0. Move the constant: x² + (b/a)x = −(c/a). Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b/2a)² − c/a. The left side is now (x + b/2a)². Take the square root of both sides. This technique converts a general quadratic into vertex form and is the algebraic proof behind the quadratic formula itself.

Real-world quadratics model projectile motion, area optimization, and break-even analysis. Projectile: height h(t) = −16t² + 80t + 6 (feet, seconds). When does it land? Set h = 0: 16t² − 80t − 6 = 0. Discriminant = 6400 + 384 = 6784. t = (80 + √6784)/32 ≈ (80 + 82.4)/32 ≈ 5.075 seconds. Maximum height at vertex t = 80/(2×16) = 2.5 seconds: h(2.5) = −100 + 200 + 6 = 106 feet.