Quadratic Equations Roots Calculator

A quadratic equation is a second-degree polynomial equation in a single variable. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation.

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

ax² + bx + c = 0

Where:

a, b, and c are constants
a ≠ 0 (if a = 0, the equation becomes linear)
x is the variable

Quadratic equations can be solved using various methods, including factoring, completing the square, and using the quadratic formula.

How to solve quadratic equations

There are several methods to solve quadratic equations:

Factoring: Express the quadratic as a product of two binomials.
Completing the square: Rewrite the equation in the form (x + p)² = q.
Quadratic formula: Use the formula x = [-b ± √(b² - 4ac)] / (2a).

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

Quadratic formula

The quadratic formula provides the roots of any quadratic equation in the form ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are the coefficients from the quadratic equation
√(b² - 4ac) is the discriminant
The ± symbol indicates there are two roots

The discriminant (b² - 4ac) 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 conjugate roots

Example calculation

Let's solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.

Given:

a = 1
b = -5
c = 6

First, calculate the discriminant:

b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

Since the discriminant is positive, there are two distinct real roots.

Now apply the quadratic formula:

x = [5 ± √1] / 2

This gives two solutions:

x = (5 + 1)/2 = 3
x = (5 - 1)/2 = 2

Therefore, the roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Frequently Asked Questions

What is the difference between a linear and quadratic equation?

A linear equation has a degree of 1 (e.g., y = mx + b), while a quadratic equation has a degree of 2 (e.g., y = ax² + bx + c). Quadratic equations can have parabolas as their graphs, while linear equations have straight lines.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is positive, there are two distinct real roots. If it's zero, there's exactly one real root.

Can quadratic equations have complex roots?

Yes, if the discriminant (b² - 4ac) is negative, the quadratic equation will have two complex conjugate roots. These roots are not real numbers but can be expressed in the form a + bi, where i is the imaginary unit.

What is the vertex of a quadratic equation?

The vertex of a quadratic equation is the point where the parabola represented by the equation reaches its maximum or minimum value. For a quadratic equation in the form y = ax² + bx + c, the x-coordinate of the vertex is given by -b/(2a).

How can I graph a quadratic equation?

To graph a quadratic equation, you can use the vertex form (y = a(x - h)² + k) to identify the vertex (h, k), then plot points around the vertex to draw the parabola. Alternatively, you can find the roots and y-intercept, then plot these points and draw the parabola through them.