M7MBA wrote: ↑

Wed Oct 07, 2020 10:14 am

\(x\) 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

Answer: 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. Since13^2 + 15^2 = 169 + 225 = 394, so x = 13 and y = 15 and xy is therefore 195.

Answer: B