Determine the solution set for the following

(1 + 6/x^2)^2 - 11(1 + 6/x^2)+28 = 0

Show complete solution and explain the answer.

### 1 Answer | Add Yours

Let n = (1+6/x^2)

n^2 - 11n + 28 = 0

Factor

(n - 7)(n - 4) = 0

Solve

n = 7

n = 4

Substitute (1+6/x^2) back in for n

n = 7

1+6/x^2 = 7

Subtract 1 from both sides

6/x^2 = 6

Multiply both sides by x^2

6 = 6x^2

Divide both sides by 6

1 = x^2

x = 1 or -1

Now repeat this process when n = 4.

n = 4

1+6/x^2 = 4

Subtract 1 from both sides

6/x^2 = 3

Multiply both sides by x^2

6 = 3x^2

Divide both sides by 3

2 = x^2

x = sqrt(2) or -sqrt(2)

**Solution set: {-sqrt(2), -1, 1, sqrt(2)}**

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