Question

If the simplified form ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}$ is $3\left( {{p}^{2}}-{{q}^{2}} \right)$ then enter 1 else 0.

Answer 1

Hint: To solve this question, we will assume variables S and R to ${{\left( 2.5p-1.5q \right)}^{2}}\text{ and }{{\left( 1.5p-2.5q \right)}^{2}}$ then expand both S and R using formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ both separately and finally subtract S and R to get our result and match is it $3\left( {{p}^{2}}-{{q}^{2}} \right)$ or not.

Complete step by step answer:
We have to simplify the term given as,
\[{{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}\]
Let, \[S={{\left( 2.5p-1.5q \right)}^{2}}\text{ and }R={{\left( 1.5p-2.5q \right)}^{2}}\]
Let us simplify term \[S={{\left( 2.5p-1.5q \right)}^{2}}\]
We will expand S using the formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
Substitute a = 2.5p and b = 1.5q and using above formula we get:
\[\begin{align}
  & S={{\left( 2.5p-1.5q \right)}^{2}} \\
 & \Rightarrow {{\left( 2.5p \right)}^{2}}-2\times \left( 2.5p \right)\left( 1.5q \right)+{{\left( 1.5q \right)}^{2}} \\
 & \Rightarrow 6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)} \\
\end{align}\]
Similarly, we will expand \[R={{\left( 1.5p-2.5q \right)}^{2}}\]
Substitute a = 1.5 and b = 2.5 and using above stated formula we get:
\[\begin{align}
  & R={{\left( 1.5p-2.5q \right)}^{2}} \\
 & \Rightarrow {{\left( 1.5p \right)}^{2}}-2\times \left( 1.5p \right)\left( 2.5q \right)+{{\left( 2.5q \right)}^{2}} \\
 & \Rightarrow 2.25{{p}^{2}}-7.5pq+6.25{{q}^{2}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)} \\
\end{align}\]
So, we have to calculate S-R, subtracting equation (i) and equation (ii), we get:
\[\begin{align}
  & S-R=6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-\left( 2.25{{p}^{2}}-7.5pq+6.25{{q}^{2}} \right) \\
 & S-R=6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-2.25{{p}^{2}}+7.5pq-6.25{{q}^{2}} \\
 & S-R=6.25{{p}^{2}}-2.25{{p}^{2}}+2.25{{q}^{2}}-6.25{{q}^{2}} \\
 & S-R=4.25{{p}^{2}}+\left( -4.25 \right){{q}^{2}} \\
 & S-R=4.25\left( {{p}^{2}}-{{q}^{2}} \right) \\
\end{align}\]
So, the simplified value of ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}$ is \[4.25\left( {{p}^{2}}-{{q}^{2}} \right)\ne 3\left( {{p}^{2}}-{{q}^{2}} \right)\]
So, we have the final answer as 0.

Note:
 Always be clear that, the answer to this question is either 1 or 0 and not $3\left( {{p}^{2}}-{{q}^{2}} \right)\text{ and }4.25\left( {{p}^{2}}-{{q}^{2}} \right)$ Because we are asked to enter 1 or 0 as answer so, we have to focus on answer matching to $3\left( {{p}^{2}}-{{q}^{2}} \right)$ then it would be 1 else 0. Students can also solve the question using identity ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to the given expression directly. Then they can open the bracket and multiply the terms. Simplify further to check if it matches with given expression or not and give the final answer.