Square Equation Roots Calculator

A square equation, also known as a quadratic equation, is a second-degree polynomial equation in the form ax² + bx + c = 0. This calculator helps you find the roots of such equations by solving for x using the quadratic formula.

What is a square equation?

A square equation is a type of quadratic equation that can be written in the standard form:

Standard Form

ax² + bx + c = 0

Where:

a, b, and c are constants
a ≠ 0 (if a = 0, it's a linear equation)
x is the variable we're solving for

Square equations are fundamental in algebra and have applications in physics, engineering, and many other fields. They can have two real roots, one real root (a repeated root), or no real roots at all.

How to solve a square equation

There are several methods to solve quadratic equations:

Factoring
Completing the square
Quadratic formula

The quadratic formula is the most general method and works for all quadratic equations. The formula is:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Where the discriminant (D) is:

D = b² - 4ac

The discriminant tells us about the nature of the roots:

If D > 0: Two distinct real roots
If D = 0: One real root (repeated)
If D < 0: No real roots (complex roots)

Discriminant explained

The discriminant is a crucial part of the quadratic formula. It's calculated as:

Discriminant Formula

D = b² - 4ac

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

Positive discriminant: Two different real roots
Zero discriminant: One real root (the parabola touches the x-axis at its vertex)
Negative discriminant: No real roots (the parabola doesn't intersect the x-axis)

Note

When the discriminant is negative, the roots are complex numbers. This calculator only shows real roots.

Example calculations

Let's solve a few example quadratic equations using our calculator.

Example 1: x² - 5x + 6 = 0

Using the quadratic formula:

Calculation

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

D = (-5)² - 4(1)(6) = 25 - 24 = 1

x = [5 ± √1] / 2

Roots: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2

Example 2: 2x² + 4x + 2 = 0

Using the quadratic formula:

Calculation

a = 2, b = 4, c = 2

D = 4² - 4(2)(2) = 16 - 16 = 0

x = [-4 ± √0] / 4 = -4/4 = -1

Root: x = -1 (double root)

Frequently Asked Questions

What is the difference between a square equation and a linear equation?: A linear equation has a highest power of 1 for the variable (e.g., y = mx + b), while a square equation has a highest power of 2 (e.g., ax² + bx + c = 0).
Can a square equation have no real roots?: Yes, when the discriminant (b² - 4ac) is negative, the equation has no real roots but two complex roots.
What is the vertex of a parabola represented by a square equation?: The vertex is at (-b/2a, f(-b/2a)) where f(x) = ax² + bx + c. It's the maximum or minimum point of the parabola.
How do I know if a quadratic equation is factorable?: An equation is factorable if it can be written as (dx + e)(fx + g) = 0, where d, e, f, and g are constants. This requires that the product of a and c equals the product of d and g.