4x 2 5x 12 0 Quadratic Equation Calculator

This calculator solves quadratic equations in the form of 4x² + 2x - 5x - 12 = 0. It finds the values of x that satisfy the equation using the quadratic formula.

How to Use This Calculator

Enter the coefficients for the quadratic equation in the form ax² + bx + c = 0. The calculator will solve for x using the quadratic formula.

Note

The equation must be in standard form with all terms on one side. Combine like terms before entering the coefficients.

Quadratic Equation Basics

A quadratic equation is a second-degree polynomial equation in the form:

Standard Form

ax² + bx + c = 0

Where:

a, b, and c are coefficients
x is the variable
a ≠ 0 (otherwise it's not quadratic)

The solutions to a quadratic equation are called roots. There can be two real roots, one real root (a repeated root), or two complex roots.

Solving the Equation

The quadratic formula is used to solve for x:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) 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

Worked Example

Let's solve 4x² + 2x - 5x - 12 = 0:

Combine like terms: 4x² - 3x - 12 = 0
Identify coefficients: a = 4, b = -3, c = -12
Calculate discriminant: (-3)² - 4(4)(-12) = 9 + 192 = 201
Apply quadratic formula:
- x = [3 ± √201] / 8
- x ≈ (3 + 14.177)/8 ≈ 2.147
- x ≈ (3 - 14.177)/8 ≈ -1.522

The solutions are approximately x ≈ 2.147 and x ≈ -1.522.

Frequently Asked Questions

What is a quadratic equation?: A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0.
How do I solve a quadratic equation?: You can solve quadratic equations by factoring, completing the square, or using the quadratic formula.
What is the discriminant?: The discriminant is the part under the square root in the quadratic formula (b² - 4ac). It determines the nature of the roots.
Can quadratic equations have complex solutions?: Yes, when the discriminant is negative, the solutions will be complex numbers.
What if a = 0 in the equation?: If a = 0, the equation is no longer quadratic and should be solved as a linear equation (bx + c = 0).