Question

Expand the algebraic expression \[{{\left( p+q \right)}^{2}}\].

Answer 1

Hint: In this problem, we have to expand the given algebraic whole square formula. We can first separate the whole square format into two factors, we can then multiply the two factors using the FOIL method. We will get an equation, which is the expansion of the given algebraic whole square formula.

Complete step by step solution:
We know that the given algebraic whole square expression is,
\[{{\left( p+q \right)}^{2}}\]
We can now write the above whole square as,
\[\left( p+q \right)\left( p+q \right)\]
We can now use the method FOIL, to multiply the above two factors.
We should know that FOIL means
First Outer Inner Last, where we will first multiply the first two terms, then we will multiply the outer two terms, then we will multiply the inner two terms and we will multiply the last remaining term.
First \[\Rightarrow p\times p={{p}^{2}}\]
Outer \[\Rightarrow p\times q=pq\]
Inner \[\Rightarrow q\times p=qp\]
Last \[\Rightarrow q\times q={{q}^{2}}\]
We can now add the terms, we get
\[\Rightarrow {{p}^{2}}+pq+pq+{{q}^{2}}\]
We can see that we have similar terms in the above, which can be added, we get
\[\Rightarrow {{p}^{2}}+2qp+{{q}^{2}}\]
Therefore, the expansion of \[{{\left( p+q \right)}^{2}}\] is \[{{p}^{2}}+2qp+{{q}^{2}}\].
Therefore, \[{{\left( p+q \right)}^{2}}={{p}^{2}}+2qp+{{q}^{2}}\].

Note: Students make mistakes while using the FOIL method, where FOIL means First Outer Inner Last, where we will first multiply the first two terms, then we will multiply the outer two terms, then we will multiply the inner two terms and we will multiply the last remaining term. We can also find the expansion of the difference of the whole square with the same method.