Quadratic Formula Calculator Solve for All Real Solutions

The quadratic formula is a fundamental tool in algebra for solving quadratic equations of the form ax² + bx + c = 0. This calculator helps you find all real solutions to any quadratic equation by applying the quadratic formula.

What is the Quadratic Formula?

The quadratic formula is derived from completing the square and provides a straightforward method to find the roots of any quadratic equation. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

a, b, and c are coefficients
a cannot be zero (otherwise it's not quadratic)
x represents the variable we're solving for

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / (2a)

This formula gives two solutions for x, which may be real or complex numbers depending on the discriminant (b² - 4ac).

Note: The discriminant determines the nature of the roots:

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

How to Use This Calculator

Using our quadratic formula calculator is simple:

Enter the coefficients a, b, and c in the input fields
Click the "Calculate" button
View the solutions in the results panel
See the graphical representation of the quadratic function

The calculator will:

Calculate both solutions using the quadratic formula
Show the discriminant value
Display the nature of the roots
Generate a chart of the quadratic function

Tip: For best results, ensure a is not zero and that the numbers are properly formatted. The calculator handles all real number inputs.

Quadratic Formula Examples

Let's look at some examples of how to solve quadratic equations using the quadratic formula.

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0

a = 1, b = -5, c = 6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1

Solutions:

x = [5 ± √1] / 2

x₁ = (5 + 1)/2 = 3

x₂ = (5 - 1)/2 = 2

Example 2: One Real Root (Repeated)

Solve 2x² - 4x + 2 = 0

a = 2, b = -4, c = 2

Discriminant = (-4)² - 4(2)(2) = 16 - 16 = 0

Solution:

x = [4 ± √0] / 4 = 4/4 = 1

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

a = 1, b = 2, c = 5

Discriminant = 2² - 4(1)(5) = 4 - 20 = -16

Solutions:

x = [-2 ± √-16] / 2 = [-2 ± 4i] / 2

x₁ = -1 + 2i

x₂ = -1 - 2i

Quadratic Formula FAQ

What is the quadratic formula used for?

The quadratic formula is used to find the roots of any quadratic equation. It's particularly useful when factoring is difficult or when dealing with equations that don't factor neatly.

How do I know if an equation has real solutions?

An equation has real solutions if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the solutions will be complex numbers.

Can the quadratic formula be used for all quadratic equations?

Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient 'a' is not zero. It's a universal method for solving quadratic equations.

What does the discriminant tell me about the solutions?

The discriminant provides information about the nature of the roots:

Positive discriminant: Two distinct real roots
Zero discriminant: One real root (repeated)
Negative discriminant: Two complex roots