Problem 118 - College Algebra

Question

What is a quadratic equation?

Step-by-Step Solution

Answer

A quadratic equation is a second order polynomial equation in a single variable. It follows the general form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants and \(x\) is an unknown variable. It is solved using the quadratic formula and the nature of its roots depends upon the discriminant \((b^2 -4ac)\).

Step 1: Define Quadratic Equation

A quadratic equation is a second order polynomial equation in a single variable. It is a type of polynomial equation of degree 2, which means it has the highest power of 2. The general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is an unknown variable.

Step 2: Note the Characteristics

In the equation \(ax^2 + bx + c = 0\), \(a\) is the coefficient of the term with the highest degree (the term with the variable \(x\) raised to the power of 2), \(b\) is the coefficient of the second term (the term with the variable \(x\) raised to the power of 1) and \(c\) is the constant term. Also, for a quadratic equation, \(a\) cannot be equal to 0 because if \(a\) were 0, the equation would become linear, not quadratic.

Step 3: Explain Solutions of Quadratic Equation

The solutions to a quadratic equation are given by the quadratic formula: \[x = \frac{{-b \pm \sqrt{{b^2 -4ac}}}}{{2a}}\]. The expression \(b^2 -4ac\) inside the square root is called the discriminant and it determines the nature of the roots of the quadratic equation. If the discriminant is positive, there are two distinct real roots. If the discriminant is zero, there is one real root (or a repeated root). If the discriminant is negative, there are two complex roots.