Systems with Distinct Roots Calculator

What are Distinct Roots?

Distinct roots refer to the two different real solutions of a quadratic equation. A quadratic equation in the standard form is:

For a quadratic equation to have two distinct real roots, the discriminant must be positive. The discriminant (D) is calculated as:

When D > 0, the equation has two distinct real roots. These roots can be found using the quadratic formula:

How to Calculate Systems with Distinct Roots

To find the distinct roots of a quadratic equation, follow these steps:

1. Identify the coefficients a, b, and c in the equation ax² + bx + c = 0.
2. Calculate the discriminant using D = b² - 4ac.
3. If D > 0, the equation has two distinct real roots.
4. Use the quadratic formula to find the roots.

The Formula

The roots of a quadratic equation ax² + bx + c = 0 are given by:

Where:

• x₁ and x₂ are the two distinct real roots
• a, b, and c are the coefficients of the quadratic equation
• √(b² - 4ac) is the square root of the discriminant

Worked Example

Let's find the distinct roots of the equation 2x² - 5x - 3 = 0.

1. Identify the coefficients: a = 2, b = -5, c = -3.
2. Calculate the discriminant: D = (-5)² - 4(2)(-3) = 25 + 24 = 49.
3. Since D = 49 > 0, there are two distinct real roots.
4. Apply the quadratic formula:
   • x₁ = [5 - √49] / 4 = (5 - 7)/4 = -2/4 = -0.5
   • x₂ = [5 + √49] / 4 = (5 + 7)/4 = 12/4 = 3

The distinct roots are x = -0.5 and x = 3.

Interpreting the Results

The roots of a quadratic equation represent the points where the parabola intersects the x-axis. For a quadratic equation with distinct real roots:

• The parabola crosses the x-axis at two distinct points.
• The vertex of the parabola lies between the two roots.
• The equation has two distinct real solutions.

Understanding the roots helps in analyzing the behavior of the quadratic function and solving real-world problems involving quadratic relationships.