Question

Find the product of the expression \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)\].

Answer 1

Hint: Here we need to apply the difference of two squares formula that is given as: \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]. So here compare and b with the given expression and then find the product accordingly.

Complete step-by-step answer:
In the question, we have to find the product of the expression \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)\]. So here we can apply the formula \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]. Now here a and b can be a variable or can also be a constant, or even a mix of both variable and constant. The value of the expression will be finite and measurable.
So, now we will compare the given expression \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)\]with \[\left( a+b \right)\left( a-b \right)\]. This will give us \[a=y,b=9\]. So here one is a constant and another is a variable. Still the formula \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\] can be applied here in order to find the product of the expression \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)\]. Now, we are just left will applying the formula \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]using the values of \[a=y,b=9\]. So, we get following:
\[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)={{y}^{2}}-{{9}^{2}}\]. Now, it is known that \[{{9}^{2}}=81\], so we can further simplify the expression as follows:
\[{{y}^{2}}-{{9}^{2}}={{y}^{2}}-81\]
Hence, we can finally say that \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)={{y}^{2}}-81\], and hence this is the final product of the expression \[\left( y\text{ }+\text{ }9 \right)\left( y-9 \right)\].

Note: We have to careful that \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]and not \[\left( a-b \right)\left( a-b \right)\ne {{a}^{2}}-{{b}^{2}}\]or
\[\left( a+b \right)\left( a+b \right)\ne {{a}^{2}}-{{b}^{2}}\]. So, this shows that only if the variable are constant are same but of opposite sign and in product form then only we can apply the formula \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\], else not.