Question

Find ${{x}^{3}}+{{y}^{3}}$, if x + y = 4 and xy = 5.

Answer 1

Hint: Use the algebraic identity given by: ${{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)$ and substitute, x = a and y = b. Substitute the values of sum of x and y and the product of x and y in the above algebraic identity to get the answer.

Complete step-by-step answer:
We have been given: x + y = 4 and xy = 5 and we have to find the value of ${{x}^{3}}+{{y}^{3}}$.
We know the algebraic identity given by: ${{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right)$.
Simplifying the above identity, we get,
${{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)$
Substituting, x = a and y = b in the above identity, we have,
${{x}^{3}}+{{y}^{3}}={{\left( x+y \right)}^{3}}-3xy\left( x+y \right)$
Now, substituting the value of sum of x and y, that is x + y = 4, and the product of x and y, that is xy = 5, in the above identity, we get,
$\begin{align}
  & {{x}^{3}}+{{y}^{3}}={{\left( 4 \right)}^{3}}-3\times 5\times \left( 4 \right) \\
 & \Rightarrow {{x}^{3}}+{{y}^{3}}=64-60 \\
 & \Rightarrow {{x}^{3}}+{{y}^{3}}=4 \\
\end{align}$
Hence, the value of required expression is 4.

Note: One may note that there can be alternate methods to solve this question. We can find the values of x and y with the help of given information, by forming quadratic equations in x or y. To form a quadratic equation in x we have to substitute the value of y from the product given, in the expression of sum. Using the middle term split we will get the value of x. Now, by substituting this obtained value of x in one of the two expressions: x + y = 4 and xy = 5, we will get the value of y. Finally, we have to substitute the value of x and y in ${{x}^{3}}+{{y}^{3}}$ to get the answer.