Cal11 calculator
Search calculator...

Solve Quadratic Equation by Square Root Calculator

Reviewed by Calculator Editorial Team

This calculator solves quadratic equations using the square root method. Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants. The square root method is one of the standard techniques for solving such equations.

How to Use This Calculator

To solve a quadratic equation using this calculator:

  1. Enter the coefficients a, b, and c of your quadratic equation in the input fields.
  2. Click the "Calculate" button to compute the solutions.
  3. Review the results displayed in the result panel.
  4. Use the "Reset" button to clear the inputs and start over.

The calculator will display the solutions to the quadratic equation, including any complex roots if they exist.

Quadratic Equation Formula

The standard form of a quadratic equation is:

ax² + bx + c = 0

The solutions to this equation can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, and c are the coefficients of the quadratic equation
  • √(b² - 4ac) is the discriminant
  • The ± symbol indicates two possible solutions

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.

Step-by-Step Solution

To solve a quadratic equation using the square root method:

  1. Write the equation in standard form: ax² + bx + c = 0.
  2. Identify the coefficients a, b, and c.
  3. Calculate the discriminant: D = b² - 4ac.
  4. If D ≥ 0, compute the two real roots using the quadratic formula.
  5. If D < 0, compute the two complex roots using the quadratic formula.

This method provides a systematic approach to finding the solutions to any quadratic equation.

Worked Examples

Example 1: Real Roots

Solve the equation x² - 5x + 6 = 0.

  1. Identify coefficients: a = 1, b = -5, c = 6.
  2. Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Compute roots: x = [5 ± √1]/2 = [5 ± 1]/2.
  4. Solutions: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2.

Example 2: Complex Roots

Solve the equation x² + 2x + 5 = 0.

  1. Identify coefficients: a = 1, b = 2, c = 5.
  2. Calculate discriminant: D = 2² - 4(1)(5) = 4 - 20 = -16.
  3. Compute roots: x = [-2 ± √(-16)]/2 = [-2 ± 4i]/2.
  4. Solutions: x = -1 + 2i and x = -1 - 2i.

Frequently Asked Questions

What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form ax² + bx + c = 0.
What is the quadratic formula?
The quadratic formula is a method for solving quadratic equations, given by x = [-b ± √(b² - 4ac)] / (2a).
What is the discriminant?
The discriminant is the part of the quadratic formula under the square root, calculated as b² - 4ac. It determines the nature of the roots.
What are complex roots?
Complex roots are solutions to quadratic equations that involve the imaginary unit i, where i² = -1. They occur when the discriminant is negative.
How do I know if my quadratic equation has real roots?
Your quadratic equation has real roots if the discriminant is positive (D > 0). If the discriminant is zero, there is exactly one real root. If the discriminant is negative, there are no real roots.