Answer:
x = 4; x = -4
Explanation:
Step 1: Cross multiply and simplify:
x * x = 2(x^2 - 8)
x^2 = 2x^2 - 16
Step 1: Subtract x^2 from both sides to set up the quadratic for solving:
(x^2 = 2x^2 - 16) - x^2
0 = x^2 - 16
Step 2: Add 16 to both sides:
(0 =x^2 - 16) + 16
16 = x^2
Step 3: Take the square root of both sides:
± √16 = √(x)^2
4 = x and -4 = x
Step 4: Check for extrraneous solutions by plugging in 4 and -4 for x in the original equations:
Plugging in 4 for x in x / (x^2 - 8) = 2 / x:
4 / (4^2 - 8) = 2 / 4
4 / (16 - 8) = 1/2
4 / 8 = 1/2
1/2 = 1/2
Plugging in 4 for x in x / (x^2 - 8) = 2 / x:
-4 / ((-4)^2 - 8) = 2 / -4
-4 / (16 - 8) = -1/2
-4 / 8 = -1/2
-1/2 = -1/2
Thus, there are no extraneous solutions so we can use x = 4 and x = -4 as the solutions.