Definition. An equation that can be put into the form

with $a\ne 0$‍ is a quadratic equation. The unknown is $x$‍, and the coefficients are $a$‍, $b$‍, and $c$‍.

Because a quadratic equation has degree two, it can have at most two different solutions. These solutions may be real numbers or complex numbers. In this article, we will explore how to solve quadratic equations by a variety of methods: graphs, tables, factoring, and the quadratic formula.

Question. Is $x^2+3x=2x+6$‍ a quadratic equation? To find out, move all the terms to one side of the equation and then combine like terms.

Answer. Yes, it's a quadratic equation because we were able to put it in the form $a{x}^2+bx+c=0$‍. The coefficients are $a=1$‍, $b=1$‍, and $c=-6$‍.

One way to solve the quadratic equation $a{x}^2+bx+c=0$‍ is to graph the parabola that has the related equation $y=a{x}^2+bx+c$‍ and find the $x$‍-intercepts of the graph.

Example 1. To solve the quadratic equation $x^2+x-6=0$‍, examine the graph of the parabola with the related equation $y=x^2+x-6$‍. This graph is shown in Figure 1.

Figure 1. The graph of $y=x^2+x-6$‍ has two $x$‍-intercepts.

Since the graph crosses the $x$‍-axis at 􀀀-3 and $2$‍, the quadratic equation has two distinct solutions: x = 􀀀-3 or $x=2$‍. We can confirm this by direct substitution: substituting $x=-3$‍ into $x^2+x-6$‍ yields $(-3{)}^2+(-3)-6=9-9=0$‍, and substituting $x=2$‍ yields $2^2+2-6=6-6=0$‍.

Why does the graphing method work? If the graph crosses the $x$‍-axis at some point $(x,y)$‍, then $y$‍ must be $0$‍. Substituting $y=0$‍ into the related equation $y=a{x}^2+bx+c$‍ yields $0=a{x}^2+bx+c$‍. That’s why every $x$‍-intercept of the graph solves the original quadratic equation.

Example 2. To solve $x^2-2x+1=0$‍, examine the graph of the parabola $y=x^2-2x+1$‍ shown in Figure 2.

Figure 2. The graph of $y=x^2-2x+1$‍ has one $x$‍-intercept.

The graph touches the x-axis exactly once at $1$‍, which means the quadratic equation has only one solution: $x=1$‍. We can think of the quadratic equation as having the same root twice. You can tell when a quadratic has a “double root” because its graph is tangent to the x-axis.

Watch out! Since the $x$‍-intercept is a real number, the graphing method finds only real solutions of the quadratic equation. It will not find any complex solutions that are not real numbers. However, it will tell us whether we need to use complex numbers to solve the equation. The next example illustrates this situation.

Example 3. Does the quadratic equation $x^2-4x+5=0$‍ have any real solutions? To find out, examine the graph of the parabola $y=x^2-4x+5$‍ shown in Figure 3.

Figure 3. The graph of $y=x^2-4x+5$‍ has no $x$‍-intercepts.

Since the graph never touches the $x$‍-axis, the quadratic equation has no real solutions. This tells us that the equation has two distinct complex solutions, but it does not tell us what they are. In a later section, we will learn how to find complex solutions using the quadratic formula.

Try it yourself. Do these quadratic equations have two, one, or no real solutions? Find out by graphing their related equations in the xy-plane.