Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 and find real or complex roots.

Enter Coefficients for ax² + bx + c = 0

x² +
x +
= 0

Discriminant (Δ = b² − 4ac)

1(2 real roots)

x₁

3

x₂

2

x = [−(-5) ± √(-5² − 4×1×6)] / (2×1) = [−(-5) ± √1] / 2

Quadratic Formula Calculator Formula & How It Works

x = [−b ± √(b² − 4ac)] / 2a
  • a = coefficient of x² (a ≠ 0)
  • b = coefficient of x
  • c = constant term
  • Discriminant: Δ = b² − 4ac (Δ > 0: two real roots; Δ = 0: one root; Δ < 0: complex roots)

The quadratic formula solves any quadratic equation by completing the square analytically. The discriminant (Δ = b² − 4ac) tells you how many real roots exist. Two distinct real roots (Δ > 0); one repeated root (Δ = 0); two complex conjugate roots (Δ < 0). The parabola crosses the x-axis at the real roots.

Quadratic Formula Calculator FAQs

How do you solve a quadratic equation step by step?

Step 1: Write in standard form ax²+bx+c=0. Step 2: Identify a, b, c. Step 3: Calculate discriminant Δ=b²-4ac. Step 4: Apply x = (-b ± √Δ) / 2a. Example: x²-5x+6=0 → a=1,b=-5,c=6 → Δ=25-24=1 → x=(5±1)/2 → x=3 or x=2.

What does a negative discriminant mean?

A negative discriminant (b²-4ac < 0) means the equation has no real solutions — the parabola doesn't cross the x-axis. The solutions are complex conjugate numbers: x = (-b ± i√|Δ|) / 2a.

Can I solve a quadratic by factoring instead?

Yes, if the equation factors neatly. x²-5x+6=0 factors as (x-2)(x-3)=0, giving x=2 or x=3. The quadratic formula always works, even when factoring is difficult or impossible.

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