Solve The Following Equation Using The Quadratic Formula Calculator

The quadratic formula is a fundamental tool in algebra for solving quadratic equations of the form ax² + bx + c = 0. This calculator helps you solve such equations quickly and accurately by applying the quadratic formula.

What is the quadratic formula?

The quadratic formula is a standard method for finding the roots of a quadratic equation. A quadratic equation is any equation that can be written in the form:

Quadratic Equation

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0.

The quadratic formula provides the solutions to this equation:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

This formula gives two solutions, known as roots, for the quadratic equation.

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

If the discriminant is positive, there are two distinct real roots.
If the discriminant is zero, there is exactly one real root (a repeated root).
If the discriminant is negative, there are two complex conjugate roots.

How to use the quadratic formula calculator

Using the quadratic formula calculator is straightforward:

Enter the coefficients a, b, and c from your quadratic equation.
Click the "Calculate" button to solve the equation.
Review the results, which include the roots and discriminant.
Interpret the results based on the discriminant value.

Note

The coefficient 'a' must not be zero. If a = 0, the equation is no longer quadratic and cannot be solved using this formula.

Worked example

Let's solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.

Example Equation

x² - 5x + 6 = 0

Here, a = 1, b = -5, c = 6.

Applying the quadratic formula:

Calculation Steps

1. Calculate the discriminant: b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

2. Since the discriminant is positive, there are two real roots.

3. Calculate the roots:

x = [5 ± √1] / 2

x₁ = (5 + 1)/2 = 3

x₂ = (5 - 1)/2 = 2

The solutions to the equation are x = 3 and x = 2.

How to interpret the results

When you use the quadratic formula calculator, you'll receive several key pieces of information:

Roots: The solutions to the quadratic equation.
Discriminant: Indicates the nature of the roots.
Nature of Roots: Describes whether the roots are real, repeated, or complex.

Understanding these results helps you analyze the quadratic equation and its solutions.

Frequently Asked Questions

What is the quadratic formula used for?

The quadratic formula is used to find the roots of any quadratic equation. It's a fundamental tool in algebra and has applications in various fields such as physics, engineering, and economics.

Can the quadratic formula be used for non-quadratic equations?

No, the quadratic formula can only be used for quadratic equations (degree 2). If the coefficient 'a' is zero, the equation is no longer quadratic and cannot be solved using this formula.

What does the discriminant tell us about the roots?

The discriminant (b² - 4ac) tells us the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex conjugate roots.