Question

If $x + y = 12$ and $xy = 27$, then find the value of ${x^3} + {y^3}$

Answer 1

Hint: According to the question we have to determine the value of ${x^3} + {y^3}$ if $x + y = 12$ and $xy = 27$. So, to obtain the value of ${x^3} + {y^3}$ first of all we have to use the given expression which are $x + y = 12$ and $xy = 27$.
Now, we have to use the formula to find the value of ${x^3} + {y^3}$ which is mentioned below:

Formula used: $ \Rightarrow {(x + y)^3} = {x^3} + {y^3} + 3xy(x + y)........................(A)$
Now, we just need to substitute all the values of the expressions we have $x + y = 12$ and $xy = 27$.

Complete step-by-step solution:
Step 1: First of all we have to use the formula (A) to determine the value of the given expression as mentioned in the solution hint. Hence,
$ \Rightarrow {(x + y)^3} = {x^3} + {y^3} + 3xy(x + y)........................(1)$
Step 2: Now, we have to substitute both of the given expressions $x + y = 12$and $xy = 27$ in the expression (1) as just obtained in the solution step 1.
$
   \Rightarrow {(12)^3} = {x^3} + {y^3} + 3 \times 27(12) \\
   \Rightarrow 1728 = {x^3} + {y^3} + 972 \\
   \Rightarrow {x^3} + {y^3} = 1728 - 972 \\
   \Rightarrow {x^3} + {y^3} = 756
 $

Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the value of the expression ${x^3} + {y^3} = 756$

Note: It is necessary that we have to substitute all the values in the formula (A) as mentioned in the solution hint so that we can easily determine the value of the expression.
For a cubic expression/equation we can obtain three roots but for a quadratic expression we can obtain only two possible roots.