Question

(\frac{x - 8}{x} = \frac{3}{x + 5})

Explanation:

Step1: Cross - multiply to eliminate fractions

Cross - multiplying the equation \(\frac{x - 8}{x}=\frac{3}{x + 5}\) gives \((x - 8)(x + 5)=3x\).

Step2: Expand the left - hand side

Expanding \((x - 8)(x + 5)\) using the FOIL method: \(x\times x+x\times5-8\times x - 8\times5=x^{2}+5x-8x - 40=x^{2}-3x - 40\). So the equation becomes \(x^{2}-3x - 40 = 3x\).

Step3: Rearrange into standard quadratic form

Subtract \(3x\) from both sides: \(x^{2}-3x-3x - 40=0\), which simplifies to \(x^{2}-6x - 40 = 0\).

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \(- 40\) and add up to \(-6\). The numbers are \(-10\) and \(4\). So, \(x^{2}-6x - 40=(x - 10)(x + 4)=0\).

Step5: Solve for x

Setting each factor equal to zero: \(x-10 = 0\) or \(x + 4=0\). So \(x = 10\) or \(x=-4\). We need to check for extraneous solutions by plugging back into the original equation. For \(x = 10\), \(\frac{10 - 8}{10}=\frac{2}{10}=\frac{1}{5}\) and \(\frac{3}{10 + 5}=\frac{3}{15}=\frac{1}{5}\). For \(x=-4\), \(\frac{-4 - 8}{-4}=\frac{-12}{-4}=3\) and \(\frac{3}{-4 + 5}=\frac{3}{1}=3\). Both solutions are valid.

Answer:

\(x = 10\) or \(x=-4\)