Quadratic equation solver with steps
ax² + bx + c = 0 with every step shown: discriminant, real or complex roots, vertex form, sum and product.
Result
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
- Vertex of the parabola
- (α; β) = (1.5; −0.25)
- Vertex form
- 1 × (x − 1.5)² − 0.25
- Sum and product of the roots (Vieta)
- S = −b/a = 3; P = c/a = 2
Note
Δ > 0: two distinct real solutions, x₁ = 1 and x₂ = 2.
Calculation, assumptions and sources
OverviewSolves ax² + bx + c = 0 step by step, from the discriminant Δ to the solutions: two real roots, a repeated root, or two complex conjugate roots when Δ < 0.
Solves ax² + bx + c = 0 step by step, from the discriminant Δ to the solutions: two real roots, a repeated root, or two complex conjugate roots when Δ < 0. The vertex form, the vertex of the parabola and the sum and product of the roots are also given. The case a = 0 is solved as a linear equation.
Method & formulasΔ = b² − 4ac
If Δ > 0, the parabola crosses the x-axis at two points; if Δ = 0, it touches it at its vertex; if Δ < 0, it does not cross it and the two solutions are complex conjugates.
- Discriminant (Δ)
- the quantity b² − 4ac.
- Repeated root
- the single solution obtained when Δ = 0.
- Complex conjugate roots
- the two solutions re ± im·i obtained when Δ < 0.
- Sum and product (Vieta)
- S = −b/a and P = c/a, handy to check the roots by hand.
Validity domainCoefficients are real numbers.
Coefficients are real numbers. Symbolic coefficients, complex coefficients and higher-degree equations are not handled.