Question

question
solve for all values of x,
\\(\frac{x + 2}{x + 9} = \frac{8}{x}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{x + 2}{x+9}=\frac{8}{x}\), cross - multiplying (since \(x
eq0\) and \(x
eq - 9\) to avoid division by zero) gives us \(x(x + 2)=8(x + 9)\).

Step2: Expand both sides of the equation

Expanding the left - hand side: \(x(x + 2)=x^{2}+2x\).
Expanding the right - hand side: \(8(x + 9)=8x + 72\).
So the equation becomes \(x^{2}+2x=8x + 72\).

Step3: Rearrange the equation to form a quadratic equation

Subtract \(8x\) and \(72\) from both sides to get \(x^{2}+2x-8x - 72 = 0\).
Simplify the left - hand side: \(x^{2}-6x - 72=0\).

Step4: Solve the quadratic equation

We can solve the quadratic equation \(x^{2}-6x - 72 = 0\) using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 1\), \(b=-6\), and \(c=-72\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-6)^{2}-4\times1\times(-72)=36 + 288 = 324\).
Then, \(x=\frac{-(-6)\pm\sqrt{324}}{2\times1}=\frac{6\pm18}{2}\).
We have two solutions:

• When we take the plus sign: \(x=\frac{6 + 18}{2}=\frac{24}{2}=12\).
• When we take the minus sign: \(x=\frac{6-18}{2}=\frac{-12}{2}=-6\).

We need to check these solutions in the original equation to make sure they do not make the denominators zero.
For \(x = 12\): The denominators are \(x + 9=12 + 9 = 21\) and \(x = 12\), both non - zero.
For \(x=-6\): The denominators are \(x + 9=-6 + 9 = 3\) and \(x=-6\), both non - zero.

Answer:

The solutions are \(x = 12\) and \(x=-6\)