Question

Find, if $16{x^2} + 24x + 9$ is a perfect square trinomial and also find its factors.

Answer 1

Hint:Perfect square is a number formed when an integer is multiplied by the same integer and if there are three terms in a math equation then it is said to be a trinomial. Factors are those whole numbers that are multiplied together to form another number.

Complete step by step answer:
In this problem, we have to find that $16{x^2} + 24x + 9$ is a perfect square trinomial or not and also we have to find its factors. So, to find this, let us recall an identity,
$ \Rightarrow {(a + b)^2} = {a^2} + 2ab + {b^2}$
Now, we have to check that $16{x^2} + 24x + 9$ is a perfect square trinomial, so, firstly we will try to make it like the identity shown above. We know that, $16$ is a square of 4, as $4 \times 4 = 16$ and $9$ is the square of $3$, as$3 \times 3 = 9$.Now, let us put these values in the given equation,
$ \Rightarrow {(4x)^2} + 24x + {(3)^2}$
Now, let us assume $4$ be a and $3$ be b and the middle term is $2ab$, so, if put the value of a and b in the middle term, we will find,
$ \Rightarrow 2ab = 2 \times 4x \times 3 = 24x$
and $24x$ is the middle term of the given equation. Now, we will rewrite the given equation as,
$ \Rightarrow {(4x)^2} + 2 \times 4 \times 3x + {(3)^2}$
And this equation now looks like the right hand side of the identity, so the left side will become,
$\therefore {(4x + 3)^2}$

Hence, we can say that $16{x^2} + 24x + 9$ is a perfect square trinomial, and the factors are $4x + 3$.

Note: In this equation the factor is $4x + 3$, there is no other factor in this equation because when the given factor multiplied by itself gives the perfect square trinomial equation.
$(4x + 3)(4x + 3) \\
\Rightarrow 4x(4x + 3) + 3(4x + 3) \\
\Rightarrow 4x \times 4x + 4x \times 3 + 3 \times 4x + 3 \times 3 \\
\Rightarrow 16{x^2} + 12x + 12x + 9 \\
\Rightarrow 16{x^2} + 24x + 9 $