Real Roots of Quadratic Equation Calculator

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the value of a.

Quadratic equations are fundamental in algebra and have applications in various fields including physics, engineering, and economics. They can describe the motion of objects, the shape of surfaces, and the behavior of systems over time.

Finding Real Roots

The real roots of a quadratic equation are the real values of x that satisfy the equation. A quadratic equation can have:

• Two distinct real roots
• One real root (a repeated root)
• No real roots (the roots are complex)

The number of real roots can be determined by the discriminant of the quadratic equation, which is given by the formula:

If D > 0, there are two distinct real roots. If D = 0, there is exactly one real root. If D < 0, there are no real roots.

The Quadratic Formula

The quadratic formula is a method for finding the roots of a quadratic equation. It is derived from completing the square and is given by:

Where:

• a, b, and c are the coefficients of the quadratic equation
• √(b² - 4ac) is the square root of the discriminant
• The ± symbol indicates that there are two roots

The quadratic formula can be used to find the real roots of any quadratic equation, provided that the discriminant is non-negative.

Worked Examples

Example 1: Two Distinct Real Roots

Consider the quadratic equation x² - 5x + 6 = 0.

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

First, calculate the discriminant:

Since D > 0, there are two distinct real roots.

Now, apply the quadratic formula:

This gives two roots:

• x = (5 + 1)/2 = 3
• x = (5 - 1)/2 = 2

Example 2: One Real Root

Consider the quadratic equation x² - 4x + 4 = 0.

Here, a = 1, b = -4, and c = 4.

Calculate the discriminant:

Since D = 0, there is exactly one real root.

Apply the quadratic formula:

The equation has a repeated root at x = 2.

Example 3: No Real Roots

Consider the quadratic equation x² + 2x + 5 = 0.

Here, a = 1, b = 2, and c = 5.

Calculate the discriminant:

Since D < 0, there are no real roots. The roots are complex.

Interpreting Results

When using the quadratic equation calculator, you will receive the real roots of the equation, if they exist. The interpretation of these roots depends on the context of the problem:

• In physics, the roots might represent the times at which an object reaches a certain position.
• In engineering, the roots might represent the points at which a system reaches a critical state.
• In economics, the roots might represent the break-even points for a business.

It's important to consider the discriminant when interpreting the results. If the discriminant is negative, the equation has no real roots, which might indicate that the problem has no real solution.