Quadratic equation solver with steps

Δ = b² − 4ac

Enter the three coefficients: the tool computes the discriminant and walks through every step to the solutions, two real roots, a repeated root, or complex conjugate roots when Δ < 0 (never just “no solution”). It also gives the vertex form, the vertex of the parabola, the sum and product of the roots, and honestly handles a = 0 by solving the linear equation it becomes.

Solutions

x₁ = 1; x₂ = 2

Discriminant Δ: Δ = b² − 4ac = (-3)² − 4 × 1 × 2 = 1
Steps: √Δ = 1
Steps: x = (−(-3) ± 1) ÷ (2 × 1) → x₁ = 1, x₂ = 2

Δ > 0: two distinct real solutions, x₁ = 1 and x₂ = 2.

Vertex of the parabola(α; β) = (1.5; -0.25)

Vertex form1 × (x − 1.5)² − 0.25

Sum and product of the roots (Vieta)S = −b/a = 3; P = c/a = 2

Scientific dossier

What the tool computes, what it assumes, where it stops being valid, and where its data comes from.

Method & formulasΔ = b² − 4ac

Δ = b² − 4ac

Δ > 0: x = (−b ± √Δ) / 2a

Δ = 0: x₀ = −b / 2a

Δ < 0: x = (−b ± i√(−Δ)) / 2a

vertex form: a(x − α)² + β, with α = −b/2a and β = c − b²/4a

The sign of the discriminant decides everything: positive, the parabola crosses the x-axis at two points; zero, it touches it at its vertex; negative, it never crosses. The solutions then live in the complex numbers, not on the real line. Internally the numerically stable variant of the formula is used (via q = −(b + sign(b)·√Δ)/2 then x₁ = q/a, x₂ = c/q). Avoiding the precision loss when b² is much larger than 4ac.

Discriminant (Δ): · the quantity b² − 4ac, whose sign determines the number and nature of the solutions.
Repeated root: · the single solution obtained when Δ = 0: the parabola touches the x-axis at exactly one point, its vertex.
Complex conjugate roots: · the two solutions of the form re ± im·i obtained when Δ < 0. “No real solution” does not mean “no solution”.
Sum and product (Vieta): · S = −b/a and P = c/a: the relations linking the roots directly to the coefficients, handy to check a result by hand.

Validity domainThe tool solves an equation with real numeric coefficients: no symbolic computation (literal coefficients), no higher-degree equations.

The tool solves an equation with real numeric coefficients: no symbolic computation (literal coefficients), no higher-degree equations. No complex coefficients. The a = 0 case is solved as the linear equation it really is, the tool says so instead of showing a division by zero or refusing.

Common pitfall: the sign of b in −bThe most frequent mistake is the sign: with b = −5, the formula −b ± √Δ starts with −(−5) = +5, not −5.

The most frequent mistake is the sign: with b = −5, the formula −b ± √Δ starts with −(−5) = +5, not −5. Same trap in b²: (−5)² = 25, never −25. The displayed steps write every coefficient in parentheses with its sign, exactly where the error creeps into handwritten work. Compare your draft line by line.