SAT Math Multiple Choice Question 659: Answer and Explanation
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
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
x2 - 2x - 3 = 0
(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).