Quadratic Equations

Definition

This equation

\(\begin{align}ax^2 + bx +c = 0\end{align}\)

is the general form of what’s called a second-degree or quadratic equation. A quadratic equation is any equation in which the highest power or exponent of the variable x is 2.

Examples:

\(\begin{align}x^2 + 2x +1 = 0\end{align}\)

\(\begin{align}3x^2 + 4x – 3 = 0\end{align}\)

\(\begin{align}5x^2 – 25x – 14 = 0\end{align}\)

In the first equation, a = 1, b = 2, and c = 1. In the second equation, a = 3, b = 4, and c = -3. In the third equation, a = 5, b = -25, and c = -14.

A quadratic equation doesn’t have to have the general form, but it can be put into that form. For example, suppose you have this equation:

\(\begin{align}-5x^2 + 3x = -2\end{align}\)

Then by adding 2 to each side of the equation you get:

\(\begin{align}-5x^2 + 3x + 2 = 0\end{align}\)

and then multiplying each side of the equation by -1, you get:

\(\begin{align}5x^2 – 3x -2= 0\end{align}\)

which is in the general form, with a = 5, b = -3, and c = -2.

Quadratic Formula

Every quadratic equation can be solved by simply putting the equation into the general form

\(\begin{align}ax^2 + bx +c = 0\end{align}\)

and then plugging the values for a, b, and c into this formula:

\(\begin{align} x = \frac { – b \pm \sqrt {b^2 – 4ac} } {{2a}}\end{align}\)

First example:

\(\begin{align}x^2 + 3x +1 = 0\end{align}\)

Using the quadratic formula, with a = 1, b = 3, and c = 1, we get

\(\begin{align}b^2 – 4ac = 5\end{align}\)

so the solutions are

\(\begin{align}x = \frac{-3}{2} \pm \frac{\sqrt{5}}{2}\end{align}\)

Second example:

\(\begin{align}x^2 + 4x +5 = 0\end{align}\)

In this case, a = 1, b = 4, and c = 5, so

\(\begin{align}b^2 – 4ac = -4\end{align}\)

and since

\(\begin{align}\sqrt{-4} = 2i\end{align}\)

then the solutions from the quadratic formula are:

\(\begin{align}x = -2 \pm i\end{align}\)

Third example:

\(\begin{align}x^2 + 4x +3 = 0\end{align}\)

In this example,

\(\begin{align}b^2 – 4ac = 4\end{align}\)

so therefore the solutions are

\(\begin{align}x = -2 \pm 1\end{align}\)

which computes to x = -1 and x = -3.