Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds real and complex roots.
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How to use this calculator
The discriminant b²−4ac determines the nature of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.
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Enter the coefficient a (of x²), b (of x), and c (constant).
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For example, x² − 5x + 6 = 0 uses a=1, b=−5, c=6.
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The calculator applies the quadratic formula and shows roots.
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A negative discriminant means the roots are complex (imaginary).
Frequently asked questions
What if the discriminant is negative?
The equation has no real solutions. The two roots are complex conjugates in the form p ± qi, where i = √−1. Complex roots appear in pairs when the parabola does not cross the x-axis.
How do I verify my roots are correct?
Substitute each root back into the equation. If ax² + bx + c = 0 is satisfied, the root is correct. Alternatively, the product of the roots = c/a and the sum of the roots = −b/a (Vieta's formulas).
Can I use this for equations not in standard form?
Rearrange first. For example, 2x² = 3x − 1 becomes 2x² − 3x + 1 = 0, so a=2, b=−3, c=1. Always move all terms to one side before identifying a, b, and c.
What is the vertex of the parabola?
The vertex x-coordinate is −b/(2a). Substitute back to get the y-coordinate. The vertex is the minimum (if a > 0) or maximum (if a < 0) point of the parabola y = ax² + bx + c.
Solving quadratic equations step by step
The quadratic formula
The quadratic formula x = (−b ± √(b²−4ac)) / 2a solves any equation of the form ax² + bx + c = 0. It was derived by completing the square and works universally — even when factoring is not obvious.
Three methods to solve quadratic equations
Factoring: fastest when the roots are rational integers. Completing the square: works always, useful for deriving the vertex form. Quadratic formula: the universal method. For exams, the quadratic formula is the most reliable when factoring is not obvious.
Real-world applications of quadratic equations
Quadratics model projectile motion (height vs time), area problems (find dimensions given area), break-even analysis (where revenue equals cost), and parabolic structures in engineering and architecture. Any U-shaped relationship between two quantities is likely quadratic.
Learn more from an authoritative source:
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