Quadratic
Equation

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Formula Derivation

See how x = −b ± b²−4ac2a is derived step by step.

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Solve by Formula

Identify a, b, c — then evaluate the formula step by step to find the roots.

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Step-by-step derivation

Here, ax² + bx + c = 0

or, ax² + bx = −c

or, ax² + bxa = −ca [∴ dividing both sides by a]

or, x² + bax = −ca

or, x² + 2 × x × b2a + b2a² − b2a² = −ca [∴ completing the square]

or, (x + b2a)² = −ca + (b2a)²

or, (x + b2a)² = −ca + b²4a²

or, (x + b2a)² = b²4a² − ca

or, (x + b2a)² = b² − 4ac4a²

or, x + b2a = ± b² − 4ac4a²

or, x = −b2a ± b² − 4ac2a

∴ x = −b ± b² − 4ac2a

The Quadratic Formula

x = −b ± b² − 4ac 2a

Therefore, the roots of x are −b + b²−4ac2a and −b − b²−4ac2a

Given equation

QuadraticEquation