Quadratic Formula Calculator
Solves any quadratic equation with the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. Enter the equation in standard, vertex or factored form and get both roots as decimals and in exact simplified radical form, the discriminant and what it says about the roots, the vertex, the axis of symmetry, the y-intercept and the equation rewritten in all three forms. Switch on complex roots to solve equations with a negative discriminant, and show the working step by step.
- The coefficients are real numbers and a is not 0.
- Complex roots are only shown when you allow them; otherwise a negative discriminant is reported as having no real roots.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a; Δ = b² − 4ac; vertex h = −b ÷ 2a, k = c − b² ÷ 4a
- The discriminant Δ = b² − 4ac decides the roots: Δ > 0 gives two different real roots, Δ = 0 one repeated root (the vertex touches the x-axis) and Δ < 0 two complex conjugate roots, (−b ± i√|Δ|) ÷ 2a.
- x₁ always takes the minus sign and x₂ the plus sign, so for a negative a the first root is the larger one.
- Vertex form a(x − h)² + k expands to b = −2ah and c = ah² + k, so Δ = −4ak. Factored form a(x − x₁)(x − x₂) expands to b = −a(x₁ + x₂) and c = a·x₁·x₂; a complex pair p ± qi gives b = −2ap and c = a(p² + q²).
- The exact form is shown when a, b and c have at most four decimal places: the equation is scaled to whole numbers, √Δ is split into a whole number times the smallest possible root, and the fraction is reduced. The value after ≈ keeps 5 significant figures below 1 and 3 decimal places otherwise.
- Decimal roots keep up to 6 decimal places. Numbers written inside the equations are rounded to 10 significant figures, factored-form roots to 6 decimal places and complex parts there to 4.
Solving 4x² + 4x − 3 = 0
Inputs
- Equation form: standard
- Negative discriminant: no
- Coefficient a: 4
- Coefficient b: 4
- Constant c: -3
- Step-by-step working: no
Δ = 4² − 4 × 4 × (−3) = 16 + 48 = 64, and √64 = 8. The roots are (−4 − 8) ÷ 8 = −3/2 = −1.5 and (−4 + 8) ÷ 8 = 1/2 = 0.5. The vertex is at (−0.5, −4), the parabola opens upward and crosses the y-axis at −3, and the equation factors as 4(x + 1.5)(x − 0.5) = 0.
Sources & references
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. It comes from completing the square and works for every quadratic equation, including ones that cannot be factored with whole numbers.
What does the discriminant tell me?
Δ = b² − 4ac is the part under the square root. If it is positive the parabola crosses the x-axis twice, if it is 0 it just touches it at the vertex, and if it is negative it never reaches the axis, so the equation has no real roots, only two complex ones.
How do I solve an equation with a negative discriminant?
Use √(−n) = i√n. For x² + 2x + 5 = 0, Δ = 4 − 20 = −16, so √Δ = 4i and x = (−2 ± 4i) ÷ 2 = −1 ± 2i. Choose “Allow complex roots” to get these instead of a warning.
Why can't a be 0?
Without the x² term the equation bx + c = 0 is linear. It has the single root x = −c ÷ b, and the quadratic formula would divide by zero.
How do I find the vertex of a parabola?
Its x-coordinate is h = −b ÷ 2a, halfway between the roots, and its y-coordinate is k = c − b² ÷ 4a, which is also −Δ ÷ 4a. For 4x² + 4x − 3 that gives (−0.5, −4).
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