Question

If it is given that \[p+q=5\] and \[pq=6\], then find the value of the expression\[{{p}^{3}}+{{q}^{3}}\].

Answer 1

Hint: Use the expansion formula for \[{{(p+q)}^{3}}={{p}^{3}}+{{q}^{3}}+3pq(p+q)\], and then find the required value of \[{{p}^{3}}+{{q}^{3}}\].

Complete step by step solution:
In the question, it is given that \[p+q=5\] and\[pq=6\] where we have to find the value of the expression\[{{p}^{3}}+{{q}^{3}}\].
So, here we will use the expansion formula for \[{{(p+q)}^{3}}={{p}^{3}}+{{q}^{3}}+3pq(p+q)\].
Now, we will substitute the value of \[p+q=5\] and\[pq=6\] in this cubic expansion. So we have:
\[\begin{align}
  & \Rightarrow {{p}^{3}}+{{q}^{3}}+3pq(p+q)={{(p+q)}^{3}} \\
 & \Rightarrow {{p}^{3}}+{{q}^{3}}+3\times 6\times 5\,\,={{(5)}^{3}}\,\,\,\,\,\,\,\,\,\,\,\because ~p+q=5,\,pq=6 \\
 & \Rightarrow {{p}^{3}}+{{q}^{3}}=125-(3\times 6\times 5) \\
 & \Rightarrow {{p}^{3}}+{{q}^{3}}=35 \\
\end{align}\]
Hence, we get the required value of \[{{p}^{3}}+{{q}^{3}}\] as 35.

Note: The common error can be while doing the calculations and simplification. For example, while solving make sure we don’t write \[{{p}^{3}}+{{q}^{3}}\] as \[{{(p+q)}^{3}}\] or vice versa. So the formula \[{{(p+q)}^{3}}={{p}^{3}}+{{q}^{3}}+3pq(p+q)\] has to applied carefully.