Quadratic Equation Solver - Find Roots of ax² + bx + c = 0

Quadratic Formula

For any quadratic equation ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

Understanding the Discriminant

The discriminant (b² - 4ac) determines the nature of the roots:

Positive discriminant: Two distinct real roots
Zero discriminant: One repeated real root
Negative discriminant: Two complex conjugate roots

Step-by-Step Process

Identify coefficients a, b, and c from your equation
Calculate the discriminant: D = b² - 4ac
Apply the quadratic formula: x = (-b ± √D) / 2a
Simplify to find the roots

Examples

Example 1: Two Real Roots

Solve: x² + 5x + 6 = 0

a = 1, b = 5, c = 6

D = 25 - 24 = 1

x = (-5 ± 1) / 2

x = -3 or x = -2

Example 2: One Repeated Root

Solve: x² - 6x + 9 = 0

a = 1, b = -6, c = 9

D = 36 - 36 = 0

x = 6 / 2

x = 3 (repeated)

Example 3: Complex Roots

Solve: x² + 2x + 5 = 0

a = 1, b = 2, c = 5

D = 4 - 20 = -16

x = (-2 ± 4i) / 2

x = -1 ± 2i

Example 4: Non-Monic

Solve: 2x² - 7x + 3 = 0

a = 2, b = -7, c = 3

D = 49 - 24 = 25

x = (7 ± 5) / 4

x = 3 or x = 0.5

Example 5: Negative Coefficient

Solve: -x² + 4x - 3 = 0

a = -1, b = 4, c = -3

D = 16 - 12 = 4

x = (-4 ± 2) / -2

x = 3 or x = 1

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