Ax 2 Bx C 0 Calculator

A quadratic equation is a second-degree polynomial equation in a single variable with three coefficients. The general form is ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. This calculator helps solve for the roots of quadratic equations using the quadratic formula.

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

ax² + bx + c = 0

Where:

a, b, and c are coefficients
x is the variable
a ≠ 0 (if a = 0, it becomes a linear equation)

Quadratic equations can represent many real-world situations, such as projectile motion, area calculations, and financial modeling. The solutions to quadratic equations are called roots or zeros.

How to solve a quadratic equation

There are several methods to solve quadratic equations:

Factoring
Completing the square
Quadratic formula
Graphical method

The quadratic formula is the most reliable method as it works for all quadratic equations. The calculator uses this method.

The quadratic formula

The quadratic formula provides the roots of any quadratic equation:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

x is the solution for the variable
a, b, and c are coefficients from the quadratic equation
√(b² - 4ac) is the discriminant

The discriminant determines the nature of the roots:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: Two complex roots

Example calculation

Let's solve the equation x² - 5x + 6 = 0:

Identify coefficients: a = 1, b = -5, c = 6
Calculate discriminant: b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
Apply quadratic formula: x = [5 ± √1]/2
Calculate roots: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

The solutions are x = 2 and x = 3.

Frequently Asked Questions

What is the quadratic formula used for?: The quadratic formula is used to find the roots of any quadratic equation in the form ax² + bx + c = 0.
How do I know if a quadratic equation has real solutions?: A quadratic equation has real solutions if the discriminant (b² - 4ac) is greater than or equal to zero.
Can the quadratic formula be used for all quadratic equations?: Yes, the quadratic formula can be used to solve any quadratic equation, regardless of the values of a, b, and c.
What does the discriminant tell me about the roots?: The discriminant indicates the nature of the roots: positive for two distinct real roots, zero for one real root, and negative for two complex roots.
How do I solve a quadratic equation by factoring?: Factoring involves expressing the quadratic as a product of two binomials. This method works only when the quadratic can be easily factored.