ZovaTool

Quadratic Formula Calculator

ax² + bx + c = 0

Roots

x₁
2
x₂
1
Discriminant Δ
1
Nature
Two real roots
Vertex
(1.5, -0.25)
Axis of symmetry
x = 1.5
Sum of roots (Vieta)
3
Product of roots (Vieta)
2
y-intercept
2

Step-by-step (Quadratic formula)

Step 1
Δ = b² − 4ac = (-3)² − 4·(1)·(2) = 1
Step 2
x = (−b ± √Δ) / 2a = (−(-3) ± √1) / (2·1)
Step 3
x₁ = 2, x₂ = 1
Step 4
Factored: 1(x − (2))(x − (1))

Completing the square

Step 1
Divide by a: x² + (-3)x + (2) = 0
Step 2
Move c/a: x² + (-3)x = -2
Step 3
Add (b/2a)² = 2.25 both sides
Step 4
(x + -1.5)² = 0.25

How to use the Quadratic Formula Calculator

  1. Enter coefficients a, b, c for ax²+bx+c=0.
  2. Calculator finds both roots (real or complex).
  3. Discriminant tells you if roots are real-distinct, repeated, or complex.
  4. Vertex and axis of symmetry shown for graphing.
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The quadratic formula, derived

The quadratic formula x = (−b ± √(b²−4ac))/(2a) is derived by completing the square on ax²+bx+c = 0. It works for every quadratic equation with real coefficients.

The discriminant Δ = b² − 4ac tells you everything about the roots before you compute them: Δ > 0 gives two real roots, Δ = 0 gives one repeated root, Δ < 0 gives a complex-conjugate pair.