ID: 6ce95fc8
(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)
Comment: Intro to quadratics.
Method 1: I prefer to solve most quadratics by completing the square, as opposed to using the quadratic formula, which has a lot of components and therefore increases the probability of making a mistake somewhere along the line. The goal is to isolate the "c" term in y = ax^2 + bx + c and make the "a" term 1.
2x^2 - 2 = 2x + 3
2x^2 - 2x = 5
x^2 - x = 5/2
x^2 - x + (-1/2)^2 = 5/2 + (-1/2)^2
(x - 1/2)^2 = 10/4 + 1/4
x - 1/2 = +/- √(11/4)
x = (1/2) +/- (√11)/2
Looking at the answer choices, the correct answer is (D). With practice, this method might take 30-45 seconds. (I would not write down the first two steps above, for example, and would probably stop before working out all the arithmetic.)
Method 2: There are a few ways to solve using Desmos, but probably the easiest is to perform a bit of algebraic manipulation and focus on the x-intercepts.
(A) cannot be correct, since 2 is not an x-intercept. Close enough is not good enough. You can type in answer choices (B), (C), and (D) to see which one matches one of the x-values above.
Clearly,
(D) is the correct answer.
Method 3: There is nothing wrong with using the quadratic formula. However, you could also plug in the answer choices directly to check for equivalence. This is easy to do with a calculator, as long as you use parentheses around anything you substitute.
A. 2(2)^2 - 2 = 2(2) + 3 --> 2(4) - 2 = 4 + 3 --> 8 - 2 = 7 X
B. 2(1 - √11)^2 - 2 = 2(1 - √11) + 3 --> 2(1 - 2√11 + 11) - 2 = 2 - 2√11 + 3 --> 2(12 - 2√11) - 2 = 5 - 2√11
--> 24 - 4√11 - 2 = 5 - 2√11 --> 22 - 4√11 = 5 - 2√11 --> 17 = 2√11 X (The irrational √11 would not magically transform into the rational 17 just by multiplying by 2. Alternatively, 17/2 ≠ √11.)
C. 2(1/2 + √11)^2 - 2 = 2(1/2 + √11) + 3 --> 2(1/4 + √11 + 11) - 2 = 1 + 2√11 + 3 -->
1/2 + 2√11 + 22 - 2 = 4 + 2√11 --> 20.5 + 2√11 = 4 + 2√11 X
At this point, it would be clear that (D) must be the correct answer, but for fun, how about working it out?
D. 2((1 + √11)/2)^2 - 2 = 2((1 + √11)/2) + 3 --> 2(1/4 + √11/2 + 11/4) - 2 = 1 + √11 + 3 -->
2(12/4 + √11/2) - 2 = 4 + √11 --> 2(3 + √11/2) - 2 = 4 + √11 --> 6 + √11 - 2 = 4 + √11 -->
4 + √11 = 4 + √11 √
This would be even easier to work out using approximate decimal equivalents and a regular calculator.
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