Question

If x – y = 4 and xy = 21, then \[{{x}^{3}}-{{y}^{3}}=\]______
(a) 361
(b) 316
(c) – 188
(d) None of these

Answer 1

Hint: First of all write the formula for \[{{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+{{y}^{2}}+xy \right)\]. Now, take the equation x – y = 4, square both sides, and find the value of \[{{x}^{2}}+{{y}^{2}}\] from it. Now, find the value of \[{{x}^{3}}-{{y}^{3}}\] by substituting various values in the formula.

Complete Step-by-step answer:
We are given that x – y = 4 and xy = 21. We have to find the value of \[{{x}^{3}}-{{y}^{3}}\]. We know that,
\[{{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+{{y}^{2}}+xy \right)....\left( i \right)\]
Let us see the information given in the question.
\[xy=21\]
\[x-y=4\]
By squaring both the sides of the above equation, we get,
\[{{\left( x-y \right)}^{2}}={{\left( 4 \right)}^{2}}\]
We know that \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. By using it in the above equation, we get,
\[{{x}^{2}}+{{y}^{2}}-2xy=16\]
Now, we are given that xy = 21. So, by substituting the value of xy in the above equation, we get,
\[{{x}^{2}}+{{y}^{2}}-2\left( 21 \right)=16\]
By simplifying the above equation, we get,
\[{{x}^{2}}+{{y}^{2}}-42=16\]
By adding 42 on both the sides of the above equation, we get,
\[{{x}^{2}}+{{y}^{2}}=16+42\]
So, we get,
\[{{x}^{2}}+{{y}^{2}}=58\]
Now, we will find the value of the expression,
\[E={{x}^{2}}+{{y}^{2}}+xy\]
So, by substituting the value of \[{{x}^{2}}+{{y}^{2}}=58\] in the above expression, we get,
\[E={{x}^{2}}+{{y}^{2}}+xy\]
\[E=58+xy\]
Also, we are given that xy = 21. So by substituting the value of xy in the above expression, we get,
E = 58 + 21
E = 79
So, we get \[{{x}^{2}}+{{y}^{2}}+xy=79\]
Now, by substituting the value of \[{{x}^{2}}+{{y}^{2}}+xy=79\] in the equation (i), we get,
\[{{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+{{y}^{2}}+xy \right)\]
\[{{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( 79 \right)\]
Also, we are given that x – y = 4. So, by substituting the value of (x – y) in the above equation, we get,
\[{{x}^{3}}-{{y}^{3}}=\left( 4 \right)\left( 79 \right)\]
\[{{x}^{3}}-{{y}^{3}}=316\]
So, we get the value of \[{{x}^{3}}-{{y}^{3}}=316\].

Hence, option (b) is the right answer.

Note: In this question, students can also find x and y separately by making an equation either in x or y and solving it, so that would come out to be x = 7 and y = 3. Now, students can cross-check their answers by finding \[{{x}^{3}}-{{y}^{3}}\] that is \[{{7}^{3}}-{{3}^{3}}\] and checking if it is matching with the answer as 316 or not. Students could also find the values of x and y by estimation and then verifying it by substituting them in the given equations.