BTGmoderatorDC wrote: ↑Wed Sep 30, 2020 5:38 pm
x and y are consecutive odd integers. If x^2 + y^2 = 394, what is the value of product xy?
A. 143
B. 195
C. 255
D. 283
E. 321
OA
B
Solution:
Since x and y are consecutive odd integers, we can say:
x = y + 2
x - y = 2
Squaring both sides of our equation, we have:
(x - y)^2 = 2^2
x^2 + y^2 - 2xy = 4
x^2 + y^2 = 4 + 2xy
Since x^2 + y^2 = 4 + 2xy, we can substitute 4 + 2xy for x^2 + y^2 in the equation x^2 + y^2 = 394, and we have:
4 + 2xy = 394
2xy = 390
xy = 195 (which is the product of x and y)
Alternate Solution 1:
Since x and y are consecutive integers, we can let y > x and hence y = x + 2. Substituting x + 2 for y in the equation, we have:
x^2 + (x + 2)^2 = 394
x^2 + x^2 + 4x + 4 = 394
2x^2 + 4x - 390 = 0
x^2 + 2x - 195 = 0
(x - 13)(x + 15) = 0
x = 13 or x = -15
If x = 13, y = x + 2 = 15 and hence xy = (13)(15) = 195. If x = -15, y = x + 2 = -13 and hence xy = (-15)(-13) = 195. We see that either way, xy = 195.
Alternate Solution 2:
Since the square of a positive number is equal to the square of its opposite, we can assume that both x and y are positive. Also, we can assume that y > x. Now, let’s check the given answer choices.
A. 143
We see that the only two consecutive positive odd integers that have a product of 143 are 11 and 13. However, 11^2 + 13^2 = 121 + 169 = 290 ≠ 394. So xy can’t be 143.
B. 195
We see that the only two consecutive positive odd integers that have a product of 195 are 13 and 15. However, 13^2 + 15^2 = 169 + 225 = 394. So xy must be 195.
Answer: B
