# SAT Math Multiple Choice Question 659: Answer and Explanation

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**Question: 659**

**14.** *x*^{2} - 2*ax* + *b* = 0

In the equation above, *a* and *b* are constants. If this equation is solved for *x*, there are two solutions. What is the sum of these two solutions?

- A. 2
*a* - B. -2
*a* - C.
*b* - D. -
*b*

**Correct Answer:** A

**Explanation:**

**A**

**Advanced Mathematics (solving quadratics) MEDIUM-HARD**

Recall from Chapter 9, Lesson 5, that the solutions to quadratic of the form *x*^{2} + *bx* + *c* = 0, the sum of those solutions is -*b* (the opposite of whatever the *x* coefficient is), and the product of those solutions is *c* (whatever the constant term is). In the quadratic *x*^{2} - 2*ax* + *b* = 0, the *x* coefficient is -2*a*. Since this must be the opposite of the sum of the solutions, the sum of the solutions is 2*a*.

Although using this theorem gives us a quick and easy solution, the theorem may seem a little abstract and mysterious to you. (You might want to review Lesson 5 in Chapter 9 to refresh yourself on the proof.) So, there is another way to attack this question: just choose values of *a* and *b* so that the quadratic is easy to factor. For instance, if we choose *a* = 1 and *b* = -3, we get:

*x*^{2} - 2(1)*x* - 3 = 0

Simplify:

*x*^{2} - 2*x* - 3 = 0

Factor:

(*x* - 3)(*x* + 1) = 0

Solve with the Zero Product Property:

*x* = 3 or -1

The sum of these two solutions is 3 + -1 = 2.

Now we plug *a* = 1 and *b* = -3 into the answer choices and we get (A) 2, (B) -2, (C) -3, (D) 3. Clearly, the only choice that gives the correct sum is (A).