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Quadratic Equation Has No Real Solutions Calculator

Reviewed by Calculator Editorial Team

A quadratic equation has no real solutions when its discriminant is negative. This calculator helps you determine when this occurs by analyzing the coefficients of the equation.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The solutions to this equation are called roots. A quadratic equation can have:

  • Two distinct real solutions
  • One real solution (a repeated root)
  • No real solutions (complex roots)

The nature of the roots is determined by the discriminant, which is calculated as:

discriminant = b² - 4ac

Conditions for No Real Solutions

A quadratic equation has no real solutions when the discriminant is negative:

b² - 4ac < 0

This means the parabola represented by the equation does not intersect the x-axis, and the roots are complex numbers. The calculator uses this condition to determine whether a given quadratic equation has no real solutions.

Note: If the discriminant is zero, the equation has exactly one real solution. If positive, there are two distinct real solutions.

How to Use This Calculator

  1. Enter the coefficients a, b, and c of your quadratic equation in the form ax² + bx + c = 0
  2. Click the "Calculate" button
  3. View the result showing whether the equation has no real solutions
  4. See the detailed calculation and interpretation

The calculator will show you:

  • The discriminant value
  • Whether the equation has no real solutions
  • A graphical representation of the quadratic function

Example Calculation

Let's examine the equation x² + 2x + 5 = 0:

  • a = 1
  • b = 2
  • c = 5

Calculate the discriminant:

discriminant = (2)² - 4(1)(5) = 4 - 20 = -16

Since the discriminant is negative (-16), the equation has no real solutions. The roots are complex numbers: x = -1 ± √(-16)/2 = -1 ± 2i√2.

Frequently Asked Questions

What does it mean when a quadratic equation has no real solutions?

It means the equation's graph (a parabola) does not intersect the x-axis, and the solutions are complex numbers involving the imaginary unit i.

How do I know if my quadratic equation has no real solutions?

Calculate the discriminant (b² - 4ac). If it's negative, the equation has no real solutions.

Can a quadratic equation have exactly one real solution?

Yes, when the discriminant is zero. The equation has a repeated real root.

What are complex roots in quadratic equations?

Complex roots occur when the discriminant is negative. They are solutions in the form of a + bi, where i is the imaginary unit.