x² Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Computes the discriminant, both roots (real or complex), and the vertex of the parabola — with full step-by-step working.
The Quadratic Formula Explained
How to Solve a Quadratic Equation
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1Write in Standard FormRearrange so the equation is ax² + bx + c = 0 with a ≠ 0.
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2Identify CoefficientsPick out the values of a (x² coeff.), b (x coeff.), and c (constant).
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3Compute the DiscriminantCalculate Δ = b² − 4ac. Its sign determines root type.
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4Apply the FormulaSubstitute into x = (−b ± √Δ) / 2a. Use + for x₁, − for x₂.
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5Find the Vertexx-vertex = −b/(2a); y-vertex = c − b²/(4a). Vertex is the parabola's turning point.
Example — x² − 5x + 6 = 0
Frequently Asked Questions
The quadratic formula x = (−b ± √(b²−4ac)) / (2a) gives the solutions to any quadratic equation ax² + bx + c = 0. It was derived by completing the square and works for all quadratics.
The discriminant Δ = b² − 4ac tells you the nature of the roots before you solve. Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: two complex conjugate roots.
Complex roots (a ± bi where i = √−1) occur when the parabola does not cross the x-axis. They always come in conjugate pairs. While not real numbers, complex roots are used extensively in engineering and physics.
The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its x-coordinate is −b/(2a) and y-coordinate is found by substituting back. If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
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