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Find The Real Solutions of The Following Equation Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the real solutions of quadratic equations in the form ax² + bx + c = 0. Whether you're a student learning algebra or a professional needing quick equation solving, this tool provides accurate results and explanations.

How to Use This Calculator

Using the calculator is simple:

  1. Enter the coefficients a, b, and c from your quadratic equation.
  2. Click the "Calculate" button to find the solutions.
  3. Review the results and interpretation.
  4. Use the "Reset" button to start over with a new equation.

The calculator will determine if there are real solutions and display them in a clear format. If no real solutions exist, it will indicate that the equation has no real roots.

Formula Used

The solutions to the quadratic equation ax² + bx + c = 0 are found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are the coefficients from your equation
  • √(b² - 4ac) is the discriminant
  • If the discriminant is positive, there are two real solutions
  • If the discriminant is zero, there is one real solution
  • If the discriminant is negative, there are no real solutions

Note: For the equation to have real solutions, the discriminant (b² - 4ac) must be non-negative. If it's negative, the equation has no real roots.

Worked Example

Let's solve the equation x² - 5x + 6 = 0:

  1. Identify the coefficients: a = 1, b = -5, c = 6
  2. Calculate the discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Since the discriminant is positive, there are two real solutions
  4. Apply the quadratic formula:
    x = [5 ± √1] / 2
  5. Calculate both solutions:
    x₁ = (5 + 1)/2 = 3 x₂ = (5 - 1)/2 = 2

The solutions to x² - 5x + 6 = 0 are x = 2 and x = 3.

Interpreting Results

When you use the calculator, you'll see one of three possible results:

  1. Two real solutions: The equation crosses the x-axis at two points. This occurs when the discriminant is positive.
  2. One real solution: The equation touches the x-axis at one point. This occurs when the discriminant is zero.
  3. No real solutions: The equation never crosses the x-axis. This occurs when the discriminant is negative.

The calculator provides a visual representation of the solutions using a graph when possible, helping you understand the relationship between the equation and its solutions.

Frequently Asked Questions

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.

How do I know if an equation has real solutions?

An equation has real solutions if the discriminant (b² - 4ac) is non-negative. If the discriminant is negative, there are no real solutions.

What does it mean if the discriminant is zero?

A discriminant of zero means there is exactly one real solution, and the parabola touches the x-axis at one point.

Can I use this calculator for non-integer coefficients?

Yes, the calculator accepts any real numbers for coefficients a, b, and c, including decimals and fractions.

Is there a limit to the size of numbers I can enter?

The calculator can handle very large numbers, but extremely large values might affect precision. For most practical purposes, standard numbers work well.