SAT Math Multiple Choice Question 659: Answer and Explanation

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Question: 659

14. x2 - 2ax + 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. 2a
  • B. -2a
  • 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 x2 + 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 x2 - 2ax + b = 0, the x coefficient is -2a. Since this must be the opposite of the sum of the solutions, the sum of the solutions is 2a.

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:

x2 - 2(1)x - 3 = 0

Simplify:

x2 - 2x - 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).

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