Question

Define the roots of a quadratic equation.

Short answer

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are the values of \( x \) that satisfy the equation, found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step-by-step solution

Understanding the Quadratic Equation

A quadratic equation looks like this: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are known numbers and \( x \) is the variable. Also, \( a \) must not be zero.

Understanding the Roots

The roots of a quadratic equation are the values of \( x \) that satisfy the equation. They also represent where the graph of the equation crosses the x-axis.

Using the Quadratic Formula to Find Roots

The quadratic formula is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Using this formula, you can find the roots of any quadratic equation. Plug \( a \), \( b \), and \( c \) from your equation into the quadratic formula to find \( x \). Note that depending upon the value under the square root in the formula, you can have: two real roots (if \( b^2 - 4ac > 0 \)), one real root (if \( b^2 - 4ac = 0 \)), or two complex roots (if \( b^2 - 4ac < 0 \)).