Quadratic Equation Calculator – Roots & Discriminant

Enter a, b and c, and the calculator computes the discriminant Δ = b²-4ac first because it classifies the roots before any square root is taken. For x²-2x-8=0, Δ = 4+32 = 36 > 0, so two distinct real roots exist: x = (2±6)/2 = 4 or -2. The vertex of the parabola sits at x = -b/2a = 1, i.e. the point (1, -9), useful for graphing and for min/max word problems. Vieta checks come free: the roots should sum to -b/a = 2 (indeed 4 + (-2) = 2) and multiply to c/a = -8 (4×(-2) = -8). If Δ < 0 the calculator reports the complex pair, and if Δ = 0 it flags the single double root.

1. 1

   Enter the coefficients

   Type a, b and c from ax² + bx + c = 0

2. 2

   Read the discriminant

   Δ = b²-4ac: positive → 2 real roots, zero → 1, negative → complex

3. 3

   Get both roots

   The formula x=(-b±√Δ)/2a is evaluated with full working shown

4. 4

   Use vertex & Vieta checks

   Vertex x = -b/2a, sum = -b/a, product = c/a for instant verification

• 1

  Δ = 36 is a perfect square, so x²-2x-8 factors neatly to (x-4)(x+2)

• 2

  The axis of symmetry x = -b/2a works even when roots are complex

• 3

  Product of roots c/a quickly exposes sign errors

• Computing b² as negative when b itself is negative; (-2)² = +4

• Forgetting 4ac entirely and using √(b²) alone under the root

• Mixing up the sign of c when moving terms to standard form

• Reading Δ < 0 as "no roots" — the complex pair x = (2±4i)/2 still exists