Question

How do you factor ${r^2} - 9{t^2}$?

Answer 1

Hint: We will use the fact that $\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$. Then we will replace a by r and b by 3t so that we get the required factors for the given expression ${r^2} - 9{t^2}$.

Complete step by step solution:
We are given that we are required to factor ${r^2} - 9{t^2}$.
Since we know that we have a formula given by the following expression:-
$ \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$
Replacing a by r and b by 3t in the above expression, we will then obtain the following expression with us:-
$ \Rightarrow \left( {{r^2} - {{\left( {3t} \right)}^2}} \right) = \left( {r - 3t} \right)\left( {r + 3t} \right)$
Simplifying the left hand side by calculating the squares of the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow {r^2} - 9{t^2} = \left( {r - 3t} \right)\left( {r + 3t} \right)$

Note: The students must commit to memory the following identity which we have used in the above solution given by the following expression:-
$ \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$
The students must also note that we can also prove this formula as well.
Let us consider the right hand side of this formula for once.
$ \Rightarrow $R. H. S. = (a – b) (a + b)
Now, we know that we have the fact given by the following expression with us:-
$ \Rightarrow $(a + b) (c + d) = a (c + d) + b (c + d)
Replacing b by – b, c by a and d by b in the above mentioned equation, we will then obtain the following expression with us:-
$ \Rightarrow $(a + b) (a - b) = a (a + b) - b (a + b)
Simplifying it further by using the distributive property which states that a (b + c) = ab + ac, we will then obtain the following expression with us:-
$ \Rightarrow \left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2} - ab + ab$
Clubbing the like terms in the right hand side of the above equation, we will then obtain the following expression with us:-
$ \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)$