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Real Abnd Non Real Roots Calculator

Reviewed by Calculator Editorial Team

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. The roots of a quadratic equation are the values of x that satisfy the equation. These roots can be real or non-real (complex) depending on the discriminant of the equation.

What Are Roots in a Quadratic Equation?

The roots of a quadratic equation are the solutions to the equation. For a quadratic equation in the standard form:

ax² + bx + c = 0

The roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are coefficients of the quadratic equation
  • √(b² - 4ac) is the discriminant

The discriminant 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

Real vs. Non-Real Roots

Real Roots

Real roots are solutions to the quadratic equation that are real numbers. They occur when the discriminant is positive or zero.

Example: For the equation x² - 5x + 6 = 0, the discriminant is 25 - 24 = 1 (positive), so there are two real roots: x = 2 and x = 3.

Non-Real Roots

Non-real roots, also known as complex roots, occur when the discriminant is negative. These roots are complex numbers involving the imaginary unit i (where i² = -1).

Example: For the equation x² + 4x + 5 = 0, the discriminant is 16 - 20 = -4 (negative), so the roots are complex: x = -2 ± √(-1) = -2 ± i.

How to Calculate Roots

To calculate the roots of a quadratic equation:

  1. Identify the coefficients a, b, and c in the equation ax² + bx + c = 0
  2. Calculate the discriminant using the formula b² - 4ac
  3. If the discriminant is positive or zero, use the quadratic formula to find the real roots
  4. If the discriminant is negative, express the roots in terms of the imaginary unit i

Using the calculator on this page, you can quickly determine whether the roots of your quadratic equation are real or non-real.

Worked Example

Let's solve the quadratic equation 2x² - 4x - 6 = 0:

  1. Identify the coefficients: a = 2, b = -4, c = -6
  2. Calculate the discriminant: (-4)² - 4(2)(-6) = 16 + 48 = 64
  3. Since the discriminant is positive (64), there are two real roots
  4. Apply the quadratic formula:
    x = [4 ± √64] / 4 = [4 ± 8] / 4
  5. Calculate the roots:
    • x₁ = (4 + 8)/4 = 12/4 = 3
    • x₂ = (4 - 8)/4 = -4/4 = -1

The roots of the equation 2x² - 4x - 6 = 0 are x = 3 and x = -1.

Frequently Asked Questions

What is the difference between real and non-real roots?

Real roots are actual numbers that satisfy the equation, while non-real roots are complex numbers involving the imaginary unit i. Real roots occur when the discriminant is positive or zero, while non-real roots occur when the discriminant is negative.

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

A quadratic equation has real roots if the discriminant (b² - 4ac) is positive or zero. If the discriminant is negative, the equation has non-real roots.

Can a quadratic equation have only one real root?

Yes, a quadratic equation can have exactly one real root when the discriminant is zero. This occurs when the parabola touches the x-axis at exactly one point.

What are complex roots in a quadratic equation?

Complex roots are solutions to the quadratic equation that involve the imaginary unit i. They occur when the discriminant is negative and are expressed in the form a ± bi, where a and b are real numbers.

How do I calculate the roots of a quadratic equation?

To calculate the roots, use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). If the discriminant is positive or zero, you'll get real roots. If the discriminant is negative, you'll get complex roots.