Question

Simplify the expression-${x^2} + 6x + 9 - 4{y^2}$

Answer 1

Hint: Write the equation in such a way that it changes in the form of an identity. Then use the formula of ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ to change the expanded form into the identity form to transform the equation. Then use the formula of ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ to simplify the equation.


Complete step-by-step answer:
Given function is ${x^2} + 6x + 9 - 4{y^2}$
We have to simplify it.
We have to write the function in such a way that if forma an identity
So we can write$6 = 2 \times 3$ and $9 = 3 \times 3$
Then we can write the function as-
$ \Rightarrow {x^2} + \left( {2 \times 3 \times x} \right) + \left( {3 \times 3} \right) - 4{y^2}$
We can also write it as-
$ \Rightarrow \left[ {{x^2} + \left( {2 \times 3 \times x} \right) + {3^2}} \right] - 4{y^2}$
The function inside the bracket is in the form ${a^2} + 2ab + {b^2}$ and we know that
$ \Rightarrow {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ where a=$x$ and b=$3$
So using this identity, we can write the function as-
$ \Rightarrow $ ${\left( {x + 3} \right)^2} - 4{y^2}$
Now we can write-$4{y^2} = {\left( {2y} \right)^2}$
Then we can write the function as-
$ \Rightarrow {\left( {x + 3} \right)^2} - {\left( {2y} \right)^2}$
This is in the form ${a^2} - {b^2}$ where a=$x + 3$ and b=$2y$
We know that ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$
Then on using this identity we get,
$ \Rightarrow \left( {x + 3 + 2y} \right)\left( {x + 3 - 2y} \right)$
On rearranging we get,
Answer$ \Rightarrow \left( {x + 2y + 3} \right)\left( {x - 2y + 3} \right)$

Note: One may go wrong if they try to pair up ${x^2} - 4{y^2}$ to form the identity of ${a^2} - {b^2}$ because instead of simplifying the function we will make the function more complex as then the function will become-
$ \Rightarrow \left( {{x^2} - 4{y^2}} \right) + \left( {6x + 9} \right)$
On using identity${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$, we get-
$ \Rightarrow \left( {x - 2y} \right)\left( {x + 2y} \right) + 6x + 9$
On taking $3$ common from the third and fourth term, we get-
$ \Rightarrow $ $\left( {x - 2y} \right)\left( {x + 2y} \right) + 3\left( {2x + 3} \right)$
This makes the equation more complex instead of simplifying it.