French Floyd
10/05/2023 · Primary School
Perform the indicated operations, then simplify. \( \frac{5 y}{y+6}+\frac{7 y^{2}}{y^{2}-36}-\frac{4}{y-6} \) \( \frac{12 y^{2}-34 y+24}{(y-6)(y+6)} \) \( \frac{12 y^{2}-34 y-24}{(y+6)(y-6)} \) \( \frac{7 y^{2}+5 y-4}{(y+0)(y-6)} \) \( \frac{7 y^{2}+5 y-4}{y^{2}+2 y-36} \)
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Step-by-step Solution
To perform the indicated operations and simplify the expression
\[
\frac{5y}{y+6} + \frac{7y^2}{y^2-36} - \frac{4}{y-6},
\]
we first need to find a common denominator. The denominators are \(y + 6\), \(y^2 - 36\), and \(y - 6\).
Notice that \(y^2 - 36\) can be factored as \((y - 6)(y + 6)\). Therefore, the common denominator for all three fractions is \((y - 6)(y + 6)\).
Now, we rewrite each term with the common denominator:
1. The first term:
\[
\frac{5y}{y + 6} = \frac{5y(y - 6)}{(y + 6)(y - 6)} = \frac{5y^2 - 30y}{(y - 6)(y + 6)}.
\]
2. The second term:
\[
\frac{7y^2}{y^2 - 36} = \frac{7y^2}{(y - 6)(y + 6)}.
\]
3. The third term:
\[
-\frac{4}{y - 6} = -\frac{4(y + 6)}{(y - 6)(y + 6)} = \frac{-4y - 24}{(y - 6)(y + 6)}.
\]
Now we can combine all three fractions:
\[
\frac{5y^2 - 30y + 7y^2 - 4y - 24}{(y - 6)(y + 6)}.
\]
Combining the numerators:
\[
5y^2 + 7y^2 - 30y - 4y - 24 = 12y^2 - 34y - 24.
\]
Thus, we have:
\[
\frac{12y^2 - 34y - 24}{(y - 6)(y + 6)}.
\]
Next, we can simplify the numerator \(12y^2 - 34y - 24\). We can factor it:
To factor \(12y^2 - 34y - 24\), we look for two numbers that multiply to \(12 \times -24 = -288\) and add to \(-34\). The numbers \(-48\) and \(6\) work:
\[
12y^2 - 48y + 6y - 24 = 12y(y - 4) + 6(y - 4) = (12y + 6)(y - 4).
\]
Thus, we can rewrite the expression as:
\[
\frac{(12y + 6)(y - 4)}{(y - 6)(y + 6)}.
\]
We can factor out a \(6\) from the numerator:
\[
\frac{6(2y + 1)(y - 4)}{(y - 6)(y + 6)}.
\]
This is the simplified form of the original expression.
So the final answer is:
\[
\frac{6(2y + 1)(y - 4)}{(y - 6)(y + 6)}.
\]
Quick Answer
\[
\frac{6(2y + 1)(y - 4)}{(y - 6)(y + 6)}.
\]
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