Question

Factorise \[18{x^2} + 48x + 32\]

Answer 1

Hint: In order to factorize this expression, we will rewrite the equation by writing the coefficient of ${x^2}$ and coefficient of $x$ in terms of perfect square. Then we will use the algebraic identity to resolve the expression. With the help of algebraic identity , we can easily find the factors of the given equation.
Formula Used:
We will use the algebraic identity which can be expressed as:
${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$

Complete step-by-step answer:
The coefficient of ${x^2}$ is 18.
The coefficient of $x$ is 48.
The constant term is 32.
We will take 2 as common from the whole expression \[18{x^2} + 48x + 32\].
This can be expressed as:
$ = 2\left( {9{x^2} + 24x + 16} \right)$
We know that $9{x^2}$ can be written as ${\left( {3x} \right)^2}$ , and $16$ can be written as ${4^2}$ . Also $24x$ can be written as the product of 2, 4 and $3x$ . Hence we can rewrite the above equation in the following way:
\[ = 2\left\{ {{{\left( {3x} \right)}^2} - 2 \times 3x \times 4 + {{\left( 4 \right)}^2}} \right\}\]
By using the identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ we have $3x$ for $a$ and 4 for $b$ . Hence, we can rewrite the above expression as
\[\begin{array}{l}
 = 2\left\{ {{{\left( {3x} \right)}^2} - 2 \times 3x \times 4 + {{\left( 4 \right)}^2}} \right\}\\
 = 2\left\{ {{{\left( {3x + 4} \right)}^2}} \right\}\\
 = 2\left\{ {\left( {3x + 4} \right)\left( {3x + 4} \right)} \right\}
\end{array}\]
 Hence, we have two roots of equation as
\[\begin{array}{l}
\left( {3x + 4} \right) = 0\\
x = - \dfrac{4}{3}
\end{array}\]
And \[2\left\{ {\left( {3x + 4} \right)\left( {3x + 4} \right)} \right\}\]is the factorized form of \[18{x^2} + 48x + 32\].

Note: In this question, we are rewriting the terms in the form of perfect squares. Hence we should have prior knowledge about the squares and roots of the different numbers. We are also using the algebraic identity in the above question. This type of question can also be solved by the factorization method in which we split the coefficient of into two-term in such a way that the product of these two terms, we get the number which equals the product of coefficient of and the constant term. Then we can further solve the expression by finding common terms.