Questions: Solve the following equation. [ (4-w)/(w^2-12 w+36)+(3)/(w-6)=(4)/(w-6) ] w= square

Solution Steps

To solve the given equation, we first need to simplify the expression. Notice that the denominator \(w^2 - 12w + 36\) can be factored as \((w-6)^2\). This allows us to rewrite the equation with a common denominator. After simplifying, we can solve for \(w\) by equating the numerators and solving the resulting linear equation.

The given equation is:

\[ \frac{4-w}{w^{2}-12w+36}+\frac{3}{w-6}=\frac{4}{w-6} \]

First, factor the quadratic expression in the denominator:

\[ w^2 - 12w + 36 = (w-6)^2 \]

Rewrite the equation using the factored form:

\[ \frac{4-w}{(w-6)^2} + \frac{3}{w-6} = \frac{4}{w-6} \]

Multiply each term by \((w-6)^2\) to eliminate the denominators:

\[ (4-w) + 3(w-6) = 4(w-6) \]

Expand and simplify the equation:

\[ 4 - w + 3w - 18 = 4w - 24 \]

Combine like terms:

\[ 2w - 14 = 4w - 24 \]

Rearrange to solve for \(w\):

\[ 2w - 4w = -24 + 14 \]

\[ -2w = -10 \]

Divide by \(-2\):

\[ w = 5 \]

Final Answer