Quadratic Formula Calculator

What is the Quadratic Formula?

The quadratic formula is used to find the solutions (roots) of a quadratic equation in the form ax2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0.

The Quadratic Formula

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

This formula gives two solutions:

• x1 = (-b + sqrt(discriminant)) / (2a)
• x2 = (-b - sqrt(discriminant)) / (2a)

The Discriminant

The discriminant (D = b^2 - 4ac) determines the nature of the roots:

• D > 0: Two distinct real roots
• D = 0: One repeated real root (two equal roots)
• D < 0: Two complex conjugate roots

How to Solve Quadratic Equations

1. Identify coefficients: Write the equation as ax^2 + bx + c = 0
2. Calculate discriminant: D = b^2 - 4ac
3. Apply the formula: x = (-b +/- sqrt(D)) / (2a)
4. Simplify: Calculate both roots

Example Calculation

Solve: 2x^2 + 5x - 3 = 0

a = 2, b = 5, c = -3

Step 1: Calculate discriminant
D = b^2 - 4ac
D = 5^2 - 4(2)(-3)
D = 25 + 24 = 49

Step 2: Apply quadratic formula
x = (-5 +/- sqrt(49)) / (2*2)
x = (-5 +/- 7) / 4

Step 3: Calculate both roots
x1 = (-5 + 7) / 4 = 2/4 = 0.5
x2 = (-5 - 7) / 4 = -12/4 = -3

Solutions: x = 0.5 and x = -3

Vertex of the Parabola

The quadratic equation graphs as a parabola. The vertex (highest or lowest point) is at:

x-coordinate: x = -b / (2a)
y-coordinate: y = f(-b/(2a)) = c - b^2/(4a)

Applications of Quadratic Equations

Physics

Projectile motion, free fall problems, and kinematic equations.

Engineering

Structural design, optimization problems, signal processing.

Economics

Profit maximization, cost analysis, supply and demand modeling.

Geometry

Area problems, finding dimensions, and geometric relationships.