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Two Real Roots Calculator

Reviewed by Calculator Editorial Team

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 x represents an unknown variable. When a quadratic equation has two real roots, it means there are two distinct real numbers that satisfy the equation.

What are two real roots?

Two real roots of a quadratic equation are the two distinct real numbers that satisfy the equation. For a quadratic equation in the form ax² + bx + c = 0, the roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

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 no real roots (the roots are complex).

When the discriminant is positive, the quadratic equation has two real roots, which can be calculated using the quadratic formula.

How to find two real roots

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

  1. Identify the coefficients a, b, and c in the quadratic equation ax² + bx + c = 0.
  2. Calculate the discriminant using the formula b² - 4ac.
  3. If the discriminant is positive, proceed to calculate the roots using the quadratic formula.
  4. Simplify the expression to find the two distinct real roots.

Remember that for the quadratic equation to have two real roots, the discriminant must be positive. If the discriminant is zero or negative, the equation will not have two distinct real roots.

Quadratic formula

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

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a is the coefficient of x²
  • b is the coefficient of x
  • c is the constant term

The ± symbol indicates that there are two roots, one with the positive square root and one with the negative square root.

Example calculation

Let's find the two real roots of the quadratic equation x² - 5x + 6 = 0.

Step 1: Identify the coefficients

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

Step 2: Calculate the discriminant

Discriminant = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

Step 3: Apply the quadratic formula

x = [5 ± √1] / 2

x = [5 ± 1] / 2

Step 4: Find the two roots

First root: x = (5 + 1)/2 = 6/2 = 3

Second root: x = (5 - 1)/2 = 4/2 = 2

The two real roots of the equation x² - 5x + 6 = 0 are x = 3 and x = 2.

Frequently Asked Questions

What is the difference between two real roots and complex roots?

Two real roots are distinct real numbers that satisfy the quadratic equation. Complex roots, on the other hand, are solutions that involve imaginary numbers and occur when the discriminant is negative.

How do I know if a quadratic equation has two real roots?

A quadratic equation has two real roots if the discriminant (b² - 4ac) is positive. If the discriminant is zero, there is exactly one real root, and if it's negative, there are no real roots.

Can the quadratic formula be used for any quadratic equation?

Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient of x² (a) is not zero. If a is zero, the equation is no longer quadratic.