Question

How do you multiply ${\left( {2b + 5a} \right)^2}$?

Answer 1

Hint: We can use algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$ to find the given product. So, first substitute $x = 2b$ and $y = 5a$ in the expansion of the formula for evaluating its value. Next, square $2b$ and $5a$ in the expansion. Finally, multiply $2$, $2b$ and $5a$ in the expansion, then we will get the desired product.
The square of the sum of two terms is expanded as the sum of squares of both terms and two times the product of them.
${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$

Complete step by step solution:
We have to find ${\left( {2b + 5a} \right)^2}$
We will use algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$ to find the given product.
So, substitute $x = 2b$ and $y = 5a$ in the expansion of the formula for evaluating its value.
$ \Rightarrow {\left( {2b + 5a} \right)^2} = {\left( {2b} \right)^2} + 2 \times \left( {2b} \right) \times \left( {5a} \right) + {\left( {5a} \right)^2}$
Now, square $2b$ and $5a$ in the above expansion.
$ \Rightarrow {\left( {2b + 5a} \right)^2} = 4{b^2} + 2 \times \left( {2b} \right) \times \left( {5a} \right) + 25{a^2}$
Now, multiply $2$, $8y$ and $3$ in the above expansion.
$ \Rightarrow {\left( {2b + 5a} \right)^2} = 4{b^2} + 20ab + 25{a^2}$

Hence, ${\left( {2b + 5a} \right)^2} = 4{b^2} + 25{a^2} + 20ab$.

Note: We can determine the product ${\left( {2b + 5a} \right)^2}$ also using FOIL method.
FOIL Method:
FOIL (the acronym for first, outer, inner and last) method is an efficient way of remembering how to multiply two binomials in a very organized manner.
To put this in perspective, suppose we want to multiply two arbitrary binomials, $\left( {a + b} \right)\left( {c + d} \right)$.
1. The first means that we multiply the terms which occur in the first position of each binomial.
2. The outer means that we multiply the terms which are located in both ends (outermost) of the two binomials when written side-by-side.
3. The inner means that we multiply the middle two terms of the binomials when written side-by-side.
4. The last means that we multiply the terms which occur in the last position of each binomial.
5. After obtaining the four partial products coming from the first, outer, inner and last, we simply add them together to get the final answer.$\left( {a + b} \right)\left( {c + d} \right) = a \cdot c + a \cdot d + b \cdot c + b \cdot d$
$ \Rightarrow \left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$
Given a product ${\left( {2b + 5a} \right)^2}$ can be written as $\left( {2b + 5a} \right)\left( {2b + 5a} \right)$.
So, first multiply the pair of terms coming from the first position of each binomial.
$ \Rightarrow \left( {2b + 5a} \right)\left( {2b + 5a} \right) = 4{b^2} + \_$
Now, multiply the outer terms when the two binomials are written side-by-side.
$ \Rightarrow \left( {2b + 5a} \right)\left( {2b + 5a} \right) = 4{b^2} + 10ab + \_$
Now, multiply the inner terms when the two binomials are written side-by-side.
$ \Rightarrow \left( {2b + 5a} \right)\left( {2b + 5a} \right) = 4{b^2} + 10ab + 10ab + \_$
Now, multiply the pair of terms coming from the last position of each binomial.
$ \Rightarrow \left( {2b + 5a} \right)\left( {2b + 5a} \right) = 4{b^2} + 10ab + 10ab + 25{a^2}$
Finally, simplify by combining like terms. So, we can combine the two middle terms with variable $x$.
$ \Rightarrow \left( {2b + 5a} \right)\left( {2b + 5a} \right) = 4{b^2} + 20ab + 25{a^2}$