Question

If x2 + y2 = 38 and x - y = 2, then what is x? If there are two possible answers, then enter the larger of the two.

Answer 1

`y= 3√(2) -1 \\x = 1 +3√(2)\\`

Explanation:

First we can find x from the linear equation. So:

If x-y=2 then x=2+y

After that. We can inster x value in the cuadratic equation:

`(2 + y) {}^(2) + y {}^(2) = 38`

`4 + 4y + y {}^(2) + {y}^(2) = 38`

`2y {}^(2) + 4y + 4 - 38 = 0`

`2y {}^(2) + 4y -34 = 0`

After reorganizing terms and using factorization methods:

`2 (y+1)^2 - 36 = 0\\(y+1)^2 = 18\\y + 1 = +-√(18) \\y1 = 3√(2) -1 \\x1 = 2 -1 + 3√(2)=1 +3√(2)\\y2 = -1 - 3√(2)\\x2 = 2 -1 - 3√(2)=1 -3√(2)\\`

Finally we choose the largest value for x.

`y = 3√(2) -1 \\x = 1 +3√(2)\\`

Answer 2

If x-y=2 then x=2+y

And if you insert that in the first equation


`(2 + y) {}^(2) + y {}^(2) = 38`

`4 + 4y + y {}^(2) + {y}^(2) = 38`

`2y {}^(2) + 4y - 34 = 0`

`y = - 1 + √(17)`
Or

`y = - 1 - √(17)`


The first one is bigger so the answer is


`y = - 1 - √(17)`