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Simplify the expression ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}$ .
A. $4{{p}^{2}}-4{{q}^{2}}$
B. $6.25{{p}^{2}}-4{{q}^{2}}$
C. $4{{p}^{2}}-2.25{{q}^{2}}$
D. $6.25{{p}^{2}}-2.25{{q}^{2}}$

Last updated date: 12th Jul 2024
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Hint: Here, we need to simplify the expression ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}$ into the simplest form possible. For this, we have to incorporate two properties or formulae. These are \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\] and ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ . Using the second formula, we break down the square of the terms of the expression. After that, we add or subtract the like terms and finally arrive at the simplified form of the expression.

Complete step-by-step solution:
In this question, we are supposed to simplify the given expression to get the desired answer. So, before proceeding with this, we must know the following formulae that will be used in the expression and help us to solve it.
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
So, starting with the solution, we have ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}$ and for solving it, we have to use the second property which is stated above as ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ . Applying this property to the first term of the expression, we get,
  & \Rightarrow {{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}={{\left( 2.5p \right)}^{2}}-2\left( 2.5p \right)\left( 1.5q \right)+{{\left( 1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}} \\
 & \Rightarrow {{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}=6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}} \\
Again, applying the property to the last term of the expression, we get,
  & \Rightarrow {{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}} \\
 & =6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-\left\{ {{\left( 1.5p \right)}^{2}}-2\left( 1.5p \right)\left( 2.5q \right)+{{\left( 2.5q \right)}^{2}} \right\} \\
 & \Rightarrow {{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}} \\
 & =6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-2.25{{p}^{2}}+7.5pq-6.25{{q}^{2}} \\
Simplifying the above expression by adding and subtracting the like terms, we get,
$\Rightarrow {{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}=4{{p}^{2}}-4{{q}^{2}}$
Thus, we can conclude that the given expression can be simplified to $4{{p}^{2}}-4{{q}^{2}}$ which is Option A.

Note: This problem can also be solved in another way. We will use the property ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ instead of the two properties that were used. Using this, we can write the expression as ${{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}=\left( 2.5p-1.5q+1.5p-2.5q \right)\left( 2.5p-1.5q-1.5p+2.5q \right)$ which gets simplified to $\left( 4p-4q \right)\left( p+q \right)$ . Using the reverse of the property, we can write it as $4{{p}^{2}}-4{{q}^{2}}$ , which is the same as that of the answer which was found earlier.