Algebra

Quadratic Equations

Let’s go over one type of polynomial, quadratic equations. Quadratic equations are functions that can be written in the following form:

\[f(x)=ax^2+bx+c\]

where a, b, and c are real numbers. A quadratic equation is called a parabola, and has a U shape on a graph. All of these are examples of quadratic equations, or parabolas:

As you can see, parabolas can either open down or open up. The orange and blue parabolas open up, and the green and red parabolas open down. A parabola opens up when \(a\)is positive, and open downs when \(a\) is negative. When\(a\) is \(0\) the entire first term is canceled out and it becomes a linear function.

Another thing to notice is the root(s) of an equation. The root of a quadratic equation \(f(x\) is when it hits the x-axis, or when \(f(x)=0\). A quadratic equation can have 0, 1, or 2 real roots. In the earlier image, the red and orange parabolas have 2 roots, because they cross the x-axis twice, the blue parabola has 0 roots, because it never crosses the x-axis, and the green parabola has 1 root, because it crosses the x-axis only once.

Sometimes, it’s important to find the x-values of the roots of an equation. The roots of an equation are the answer to the following expression:

\[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]