Quadratic Formula Calculator

Quadratic Formula Calculator - Multi-Tools

Quadratic Formula Calculator

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Result

Quadratic Formula

The quadratic formula is used to solve equations in the form:

ax² + bx + c = 0

The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

Discriminant

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

  • If discriminant > 0: Two real roots
  • If discriminant = 0: One real root (double root)
  • If discriminant < 0: Two complex conjugate roots

Example Problems

Example 1: x² + 5x + 6 = 0

Step 1: Identify coefficients

  • a = 1
  • b = 5
  • c = 6

Step 2: Calculate discriminant

b² - 4ac = 25 - 24 = 1

Step 3: Apply quadratic formula

x = (-5 ± √1) / 2

Solutions: x = -2 or x = -3

Example 2: 2x² - 4x + 2 = 0

Step 1: Identify coefficients

  • a = 2
  • b = -4
  • c = 2

Step 2: Calculate discriminant

b² - 4ac = 16 - 16 = 0

Step 3: Apply quadratic formula

x = (4 ± √0) / 4

Solution: x = 1 (double root)

Vertex Form

The vertex form of a quadratic equation is:

y = a(x - h)² + k

where (h, k) is the vertex of the parabola.

To convert from standard form:

  • h = -b/(2a)
  • k = c - b²/(4a)
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