It is not necessary to actually find the solutions to a quadratic equation to determine the sum of its solutions.

First, get the equation in standard formby subtractingfrom both sides:

If a quadratic equation has two distinct solutions, which we are given here, their sum is the linear coefficient. In this problem,, makingthe correct choice.

The Intermediate Value Theorem (IVT) states that if the graph of a functionis continuous on an interval, andanddiffer in sign, thenhas a zero on. Consequently, the way to answer this question is to determine the signs ofon the endpoints of the subintervals -. We can do this by substituting each value foras follows:

assumes positive values forand negative values for. By the IVT,has a zero on.

Use the zeroes of the graph to determine the matching function. Zeroes are values ofxwhere. In other words, they are points on the graph where the curve touches zero.

Visually, you can see that the curve crosses the x-axis when,, and. Therefore, you need to look for a function that will equal zero at thesexvalues.

A function with a factor ofwill equal zero when, because the factor ofwill equal zero. The matching factors for the other two zeroes,and, areand, respectively.

The answer choicehas all of these factors, but it is not the answer because it has an additional zero that would be visible on the graph. Notice it has a factor of, which results in a zero at. This additional zero that isn't present in the graph indicates that this cannot be matching function.

is the answer because it has all of the required factors and, as a result, the required zeroes, while not having additional zeroes. Notice that the constant coefficient of negative 2 does not affect where the zeroes are.

First, look at the zeroes of the graph. Zeroes are where the function touches the x-axis (i.e. values ofwhere).

The graph shows the function touching the x-axis when,, and at a value in between 1.5 and 2.

Notice all of the possible answers are already factored. Therefore, look for one with a factor of(which will makewhen), a factor ofto makewhen, and a factor which will makewhenis at a value between 1.5 and 2.

This function fills the criteria; it has anand anfactor. Additionally, the third factor,, will result inwhen, which fits the image. It also does not have any extra zeroes that would contradict the graph.