Quadratic Equation Solver finds the roots of any quadratic in standard form a x2 + b x + c = 0, shows the discriminant, vertex, axis of symmetry, intercept, factored/vertex forms when possible, and prints a step-by-step derivation. It uses the classical quadratic formula and definitions documented in Encyclopædia Britannica (quadratic equation reference).

Quadratic Equation Solver
Equation
ax2 + bx + c = 0
a ≠ 0 for a quadratic
Results
Standard form: —
Discriminant (Δ): —
Roots: —
Vertex (h, k): —
Axis of symmetry: —
Opens: —
y-intercept: —
Step-by-step
—
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How to use

  1. Type numeric values for a, b, c (with a ≠ 0 for a true quadratic).
  2. Press Solve. Use Clear All to reset and start another problem.
  3. Read the results panel and the “Step-by-step” block. Copy the steps directly into a notebook if required.

What the solver shows

  • Standard form: The exact equation assembled from your inputs.
  • Discriminant Δ = b2 − 4ac and the nature of roots:
    • Δ > 0 → two distinct real roots.
    • Δ = 0 → one real repeated root.
    • Δ < 0 → complex conjugate roots; no real solutions.
  • Roots: Exact numeric values. Complex roots appear as p ± q i.
  • Vertex and axis: h = -b/(2a), k = a h2 + b h + c; axis is x = h.
  • Opens: up if a > 0, down if a < 0.
  • y-intercept: c.
  • Vertex form: y = a(x − h)2 + k with h, k from your inputs.
  • Factored form: a(x − r1)(x − r2) when roots are real; otherwise marked “irreducible over ℝ”.
  • Step-by-step: A line-by-line derivation: compute Δ, plug into the formula, simplify to the final roots. For Δ < 0 it explicitly states “No real solutions over ℝ” and writes the complex form.

Formulas used

  • Quadratic formula: x = (-b ± √Δ) / (2a), where Δ = b2 − 4ac.
  • Vertex: (h, k) with h = -b/(2a), k = a h2 + b h + c.
  • Axis of symmetry: x = h.
  • Factoring link: if roots are real, x - r1 and x - r2 are the linear factors.

Worked examples

Two real roots: 2x2 - 5x - 3 = 0 → Δ = 25 + 24 = 49 → x = (5 ± 7)/4 → x = 3 or x = -0.5 → factors 2(x - 3)(x + 0.5).

Repeated root: x2 - 6x + 9 = 0 → Δ = 36 - 36 = 0 → x = 6/2 = 3 (double root) → vertex at (3, 0).

No real solutions: x2 + 4x + 13 = 0 → Δ = 16 - 52 = -36 → x = (-4 ± i·6)/2 = -2 ± 3i → irreducible over ℝ.

Quadratic Equation Solver Step-by-Step

Tips and pitfalls

  • Confirm the equation is in standard form before entering coefficients. Move all terms to the left side.
  • If every coefficient shares a common factor, divide first to simplify arithmetic.
  • For graph insights, check the sign of a and the vertex (h, k). The minimum/maximum occurs at x = h.
  • Units are abstract. If the problem uses units, the roots carry the same x-units as the original model.

FAQ

Does “no real solutions” mean no answer? It means the solutions are complex. Over the real numbers there is no x that satisfies the equation; over the complex numbers there are two solutions that the solver shows.

Can I enter decimals or large numbers? Yes. The solver handles integers, decimals, and scientific notation. Results display up to six decimal places or scientific notation when needed.

Why are the factors missing sometimes? Factored form is shown only when the roots are real. With complex roots the real factorization is impossible; the tool labels it “irreducible over ℝ”.

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