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Consider \[\left( x+p \right)\left( x+q \right)={{x}^{2}}+\left( p+q \right)x+pq\].
1.Put q instead of ‘p’ what do you observe?
2.Put p instead of ‘q’ what do you observe?
3.What identities did you observe in your result?

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Last updated date: 13th Jul 2024
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Answer
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Hint: In this problem, we are given an equation, here it is said to substitute the variable q instead of ‘p’ and p instead of ‘q’, if we substitute it and simplify the equation, we will get the form as like the algebraic whole square formula. We can now see what we can observe and write the observed formula in the result.

Complete step by step answer:
Here we are given an expression,
\[\left( x+p \right)\left( x+q \right)={{x}^{2}}+\left( p+q \right)x+pq\]
We can now work out the given conditions and observe it.
1.Put q instead of ‘p’ in the given equation, we get
\[\Rightarrow \left( x+q \right)\left( x+q \right)={{x}^{2}}+\left( q+q \right)x+\left( q \right)q\]
We can now simplify the above step, we get
\[\Rightarrow {{\left( x+q \right)}^{2}}={{x}^{2}}+2qx+{{q}^{2}}\]…….. (1)
2. Put p instead of ‘q’ in the given expression, we get
\[\Rightarrow \left( x+p \right)\left( x+p \right)={{x}^{2}}+\left( p+p \right)x+p\left( p \right)\]
We can now simplify the above step, we get
\[\Rightarrow {{\left( x+p \right)}^{2}}={{x}^{2}}+2px+{{p}^{2}}\]……… (3)
3. In the above two equations (1) and (2), it seems to be an algebraic whole square property, which is of the form
\[\Rightarrow {{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
Therefore, if we put p instead of ‘q’ or q instead of ‘p’, then we will get an algebraic whole square property of the form \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\].

Note: We should know some of the algebraic properties to identify these types of formulas which occur in the given problem. We should concentrate while simplifying the steps, as we have the same terms in one of the sides, we make it the whole squared form.