Question

Find factor \[3{{x}^{2}}\,-\,48\]?

Answer 1

Hint: According to the given question we have to factorize the given quadratic expression. In order to factorize the quadratic expression we will try to take a common form of the expression and then try to express it in terms of algebraic identities. Furthermore, by simplifying it we will get our required answer.

Complete Step by Step Solution:
In this question we have given, \[3{{x}^{2}}\,-\,48\] -----(1)
In order to factorize the above given expression, Now, we will take \[3\] as greatest common factor from expression \[\left( 1 \right)\] and we get, \[3\left( {{x}^{2}}-16 \right)\].
We can rewrite the above expression as follows;
\[3\left( {{\left( x \right)}^{2}}-{{\left( 4 \right)}^{2}} \right)\] ----(2)
Since we know that \[16=\,{{\left( 4 \right)}^{2}}\].
Also, we know that the algebraic identify that is the difference of squares pattern:
\[\left( {{a}^{2}}-{{b}^{2}} \right)\,=\,\left( a+b \right)\left( a-b \right)\]
We will use this pattern to continue factory the polynomial as shown below.
\[=3\left( {{\left( x \right)}^{2}}-{{\left( 4 \right)}^{2}} \right)\]
\[=3\left( x-4 \right)\left( x+4 \right)\]
As we can observe that the above expression that is completely factorizes as it terms into linear factors.
It means that these are no more quadratics in the expression.
Which means that we have completely factored the given quadratic expression.
Hence, we can conclude that
\[3{{x}^{2}}-48=3\left( x-4 \right)\left( x+4 \right)\]

Note: There are certain methods which can be used in order to factorize any expression.
• Factoring out the Greatest Common Factor (GCF).
• The sum product pattern.
• The grouping square trinomial pattern.
• The difference of square pattern.