CalcRover

Quadratic Formula Calculator

Enter a, b and c to solve the quadratic equation: the discriminant decides whether you get two roots, one, or a complex pair, and the steps walk the formula exactly the way you'd write it on a test. The vertex comes along for graphing.

x² +x += 0

The quadratic formula

Any quadratic ax² + bx + c = 0 is solved by the formula below. The discriminant D = b² − 4ac is the part under the root and tells the story before you finish: D > 0 means two real roots, D = 0 one repeated root, D < 0 a complex-conjugate pair (the parabola misses the x-axis).

The vertex −b/2a sits exactly between the roots — it's the parabola's turning point and the maximizer/minimizer in word problems (projectile peaks, profit maxima).

x = (−b ± √(b² − 4ac)) ÷ 2a
vertex: x = −b ÷ 2a

Worked examples

Example 1: x² − 5x + 6 = 0

The classic factorable quadratic.

  1. Discriminant.
    D = 25 − 24 = 1
  2. Apply the formula.
    x = (5 ± 1) ÷ 2
  3. Both roots.
    x₁ = 3, x₂ = 2

x = 2 and x = 3 (checks out: (x−2)(x−3) = x² − 5x + 6).

Example 2: when does the ball land?

Height h = −16t² + 48t + 5 (feet, seconds). When does h = 0?

  1. Coefficients.
    a = −16, b = 48, c = 5
  2. Discriminant and formula.
    D = 2304 + 320 = 2624; t = (−48 ± 51.2) ÷ −32
  3. Keep the positive time.
    t ≈ 3.1 s

The ball lands after about 3.1 seconds (the negative root is before launch).

Example 3: no real solutions

x² + 2x + 5 = 0.

  1. Discriminant is negative.
    D = 4 − 20 = −16
  2. Complex pair.
    x = (−2 ± 4i) ÷ 2 = −1 ± 2i

No real roots — the parabola stays above the axis; complex roots −1 ± 2i.

FAQ

When should I factor instead of using the formula?+

Factor when the roots are obviously whole numbers (small D that's a perfect square). The formula always works, factoring is faster when it works — the discriminant tells you which case you're in.

What does the discriminant actually tell me?+

The number of x-axis crossings: D > 0 two, D = 0 one (tangent), D < 0 none. Its being a perfect square also means the quadratic factors over the integers.

What if a is negative?+

Nothing changes in the formula — just keep the sign through −b and 2a. The parabola opens downward, so the vertex is a maximum.

What are complex roots good for?+

They mean "no real solution" for geometry problems, but in physics and engineering complex pairs describe oscillation and damping — they're an answer, not a failure.

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