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Solve The Following Quadratic Equation Calculator

Reviewed by Calculator Editorial Team

Quadratic equations are fundamental in algebra and appear in many real-world problems. This calculator helps you solve quadratic equations of the form ax² + bx + c = 0, finding the roots using the quadratic formula. Learn how to interpret the results and understand the nature of the roots based on the discriminant.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form:

ax² + bx + c = 0

Where:

  • a, b, and c are constants
  • a ≠ 0 (if a = 0, the equation is linear, not quadratic)
  • x is the variable

Quadratic equations can represent various real-world situations, such as projectile motion, area problems, and optimization tasks. The solutions to these equations (the roots) can provide valuable insights into the problem being modeled.

The Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. The formula is derived from completing the square and is expressed as:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

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

The quadratic formula works for all quadratic equations, regardless of whether the roots are real or complex. When the discriminant is positive, there are two distinct real roots. When it's zero, there's exactly one real root (a repeated root). When it's negative, the roots are complex conjugates.

The Discriminant

The discriminant is the part of the quadratic formula under the square root, calculated as:

D = b² - 4ac

The discriminant provides important information about the nature of the roots:

  • If D > 0: Two distinct real roots
  • If D = 0: One real root (repeated)
  • If D < 0: Two complex conjugate roots

Understanding the discriminant helps you interpret the results of the quadratic equation and determine the type of solutions you're dealing with.

How to Use This Calculator

Using our quadratic equation solver is simple:

  1. Enter the coefficients a, b, and c in the input fields
  2. Click the "Calculate" button
  3. View the results, including the roots and discriminant
  4. Interpret the results based on the discriminant

The calculator will display the roots in a clear format and show the discriminant value. You can also see a visual representation of the quadratic function if you're using a modern browser.

Example Problems

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2

Solutions: x = 3 and x = 2

Example 2: One Real Root

Solve x² - 6x + 9 = 0

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

Solution: x = 3 (double root)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2 = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2

Solutions: x = -1 + 2i and x = -1 - 2i

Frequently Asked Questions

What is the difference between a linear and quadratic equation?
A linear equation has a single variable with the highest power of 1, while a quadratic equation has a variable with the highest power of 2. The quadratic equation can have two solutions, while the linear equation has only one.
When should I use the quadratic formula instead of factoring?
The quadratic formula is a reliable method that works for all quadratic equations, while factoring only works when the equation can be easily factored. The quadratic formula is particularly useful when the equation doesn't factor neatly or when you want to ensure you find all possible solutions.
What does a negative discriminant mean?
A negative discriminant indicates that the quadratic equation has two complex conjugate roots. These roots are not real numbers but involve the imaginary unit i.
Can quadratic equations have more than two solutions?
No, quadratic equations can have at most two solutions (roots). The number of solutions is determined by the discriminant: two distinct real solutions if D > 0, one real solution if D = 0, and two complex solutions if D < 0.
How can I verify the solutions to a quadratic equation?
You can verify the solutions by substituting them back into the original equation. If the equation holds true for both solutions, they are correct. Additionally, you can check the discriminant to ensure it matches the nature of the solutions.