Question

2√x - 14 = 288/√x

Explanation:

Step1: Let \( t = \sqrt{x} \) (where \( t>0 \) since square root is non - negative and \( t

eq0 \) to avoid division by zero)
The equation \( 2\sqrt{x}-14=\frac{288}{\sqrt{x}} \) becomes \( 2t - 14=\frac{288}{t} \)

Step2: Multiply both sides by \( t \) to eliminate the denominator

\( t(2t - 14)=288 \)
Expand the left - hand side: \( 2t^{2}-14t = 288 \)

Step3: Divide the entire equation by 2 to simplify

\( t^{2}-7t = 144 \)

Step4: Rearrange the equation into standard quadratic form \( ax^{2}+bx + c = 0 \)

\( t^{2}-7t - 144 = 0 \)

Step5: Factor the quadratic equation

We need two numbers that multiply to \( - 144 \) and add up to \( - 7 \). The numbers are \( - 16 \) and \( 9 \).
So, \( t^{2}-7t - 144=(t - 16)(t + 9)=0 \)

Step6: Solve for \( t \)

Set each factor equal to zero:
\( t - 16 = 0\) or \( t + 9 = 0 \)
Since \( t=\sqrt{x}>0 \), we discard \( t=-9 \). So, \( t = 16 \)

Step7: Substitute back \( t=\sqrt{x} \) to find \( x \)

If \( t = 16 \), then \( \sqrt{x}=16 \)
Square both sides: \( x = 16^{2}=256 \)

Answer:

\( x = 256 \)