Question

solve the equation by making an appropriate substitution.
\\(\left(x - \dfrac{8}{x}\
ight)^2 - 5\left(x - \dfrac{8}{x}\
ight) - 14 = 0\\)

Explanation:

Step1: Let \( u = x - \frac{8}{x} \)

Substitute \( u \) into the equation, we get \( u^2 - 5u - 14 = 0 \)

Step2: Solve the quadratic equation \( u^2 - 5u - 14 = 0 \)

Factor the quadratic: \( (u - 7)(u + 2) = 0 \)
Set each factor to zero: \( u - 7 = 0 \) or \( u + 2 = 0 \)
So \( u = 7 \) or \( u = -2 \)

Step3: Substitute back \( u = x - \frac{8}{x} \)

Case 1: \( x - \frac{8}{x} = 7 \)

Multiply both sides by \( x \) (assuming \( x
eq 0 \)): \( x^2 - 8 = 7x \)
Rearrange: \( x^2 - 7x - 8 = 0 \)
Factor: \( (x - 8)(x + 1) = 0 \)
Solutions: \( x = 8 \) or \( x = -1 \)

Case 2: \( x - \frac{8}{x} = -2 \)

Multiply both sides by \( x \) (assuming \( x
eq 0 \)): \( x^2 - 8 = -2x \)
Rearrange: \( x^2 + 2x - 8 = 0 \)
Factor: \( (x + 4)(x - 2) = 0 \)
Solutions: \( x = -4 \) or \( x = 2 \)

Step4: Check for extraneous solutions

Check \( x = 8 \): \( 8 - \frac{8}{8} = 7 \), \( 7^2 - 5\times7 - 14 = 49 - 35 - 14 = 0 \), valid.
Check \( x = -1 \): \( -1 - \frac{8}{-1} = 7 \), \( 7^2 - 5\times7 - 14 = 0 \), valid.
Check \( x = -4 \): \( -4 - \frac{8}{-4} = -4 + 2 = -2 \), \( (-2)^2 - 5\times(-2) - 14 = 4 + 10 - 14 = 0 \), valid.
Check \( x = 2 \): \( 2 - \frac{8}{2} = 2 - 4 = -2 \), \( (-2)^2 - 5\times(-2) - 14 = 4 + 10 - 14 = 0 \), valid.

Answer:

The solutions are \( x = -4, -1, 2, 8 \)