a. Factor a quadratic expression to reveal the zeros of the function it defines.

Students should already be comfortable with finding the zeros of functions. In other words, finding the x values that make the mathematical expression equal to 0. For linear expressions like 5x – 10, it's relatively easy to find the zeros. All we have to do is set the expression to equal 0 and solve for x. In this case, x = 2. Duh.

But what about quadratic expressions that aren't that simple? The simplest way to find roots for higher order expressions is by factoring. We can actually demonstrate that using our linear expression. Recall that we set our linear expression equal to 0 in order to find the root. Mathematically, we had 5x – 10 = 0. Your students most likely added 10 to both sides and then divided by 5, but let's give factoring a shot.

We can factor that expression by pulling out 5 (the coefficient of x) as the GCF. That would yield the factored form 5(x – 2) = 0. We've turned our two-term expression into a one-term expression which has two factors, 5 and (x – 2). The product of those factors is zero, however. By the zero product rule, at least one of the factors must be zero. Well, obviously, 5 can never be 0. Therefore, x – 2 must be equal to zero. If we solve the equation x – 2 = 0, we get x = 2, which is the answer we got before.

So how does this work for a quadratic? When finding the zeros of a quadratic equation, the best thing to do is try and factor. We can take an expression like 6x² – 11x – 10 and turn it into (2x – 5)(3x + 2). To find the zeros of the expression, we set this factored form equal to zero, then apply the zero product rule.

In this case, both factors contain variable expressions, so we must consider that either of the factors may be equal to zero, or both may be equal to zero simultaneously. A common mistake students make is to set the variable equal to zero, but that's not right! We must set each factor equal to zero.

So we have to solve two first order equations: 2x – 5 = 0 and 3x + 2 = 0. The first one gives . The second one gives . There should be two roots, since this is a second order expression, and these are the two roots we are looking for. The equation y = 6x² – 11x – 10 has two x-intercepts! The graph of that equation looks like this:

It crosses the x-axis two times, and at the locations we calculated. This curve is called a parabola. All quadratic expressions of this type have the same basic shape.

Sometimes, quadratics have only one x-intercept and other times they don't have any real roots at all. It just means the function doesn't cross the x-axis, or that it crosses the x-axis once (at the minimum or maximum). Make sure your students don't have a panic attack when that happens.