ID: ea6d05bb
(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)
Comment: Do only the amount of work required.
Method 1: Because the question asks about the b term, we need only focus on what would result in an x after the distribution. 3x * 6 would be one such product, and -23 * 19x the other.
3x * 6 = 18x
-23 * 19x = -437x
-437x + 18x = -419x
The correct answer is -419.
Method 2: A graph would probably not help too much with this problem. A calculator could be put to better use with the arithmetic, since there are large numbers involved. The only way I can think of in which a graph would be useful is if you were intimately familiar with finding the x-value of the vertex from the standard form of a parabola, -b/2a, and you quickly identified that "a" was 57 (from 3 * 19).
The graph shows that as "x" approaches 3.67544, the parabola bottoms out at roughly -908. We can now work with our x-vertex formula:
(3.67544) = -b/(2(57))
3.67544 = -b/114
419.00016 = -b
-419.00016 = b
Because the SAT has only so many input slots, and -419 takes four of them, and also because the part after the decimal goes to the ten-thousandths before there is any nudge away from another 0, it is pretty safe to say that -419 is the correct answer.
To be clear, I would not recommend this method over the one above, method 1, but it does show how you can put a graphing calculator to use.
Method 3: If zeroing in on the x-terms is a scary proposition, feel free to work out the entire binomial distribution, whether using the FOIL method or some other method.
(3x - 23)(19x + 6)
57x^2 + 18x - 437x - 138
57x^2 - 419x - 138
If this expression is equivalent to ax^2 + bx + c, then b must be -419. This is a more textbook way of solving the question, which might take a fraction longer than method 1 outlined earlier.
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