Question

How do you solve \[3{x^2} = 48\].

Answer 1

Hint: In this question we used the formula for factor; power of ‘a’ is two minus power of ‘b’ is two. The expression of power of ‘a’ is two minus power of ‘b’ is two is called the difference of squares. And this formula is expressed as below:
\[ \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\].

Complete step by step answer:
In the question, we have the expression as,
\[3{x^2} = 48\]
As we know that \[48\] is divided by \[3\]. Hence the above equation is written as
\[ \Rightarrow {x^2} - 16 = 0\]
 The number \[16\]is the perfect square of the number\[4\]. Then the equation is written as below.
\[{x^2} - {4^2} = 0\]
In the above equation, we can add and subtract \[4x\]. By adding and subtracting \[4x\], there is no effect in the above expression.
Then,
The above expression is written as below.
\[ \Rightarrow {x^2} - {4^2} + 4x - 4x\]
Now we will rearrange the above expression,
\[ \Rightarrow {x^2} - 4x + 4x - {4^2}\]
Then we take the common, from the first two terms we take \[x\] as the common and from the last two terms we will take \[4\] as common.
Then the above expression is written as below.
\[x\left( {x - 4} \right) + 4\left( {x - 4} \right) = 0\]
By solving the above expression, the result would be as below.
\[\left( {x - 4} \right)\left( {x + 4} \right) = 0\]
Then we will get the factor of \[x\]as.
 \[ \Rightarrow x - 4 = 0\]
\[\therefore x = 4\]
\[ \Rightarrow x + 4 = 0\]
\[\therefore x = - 4\]
Then,
\[\therefore x = \pm 4\]

Therefore, the factors of \[x\] are \[ \pm 4\].

Note:
We will derive the formula for the expression as,
\[{a^2} - {b^2}\]
In the above expression, we can add and subtract the\[ab\]. By adding and subtracting the\[ab\], there is no effect in the above expression.
Then,
The above expression is written as below.
\[ \Rightarrow {a^2} - {b^2} + ab - ab\]
Now we will rearrange the above expression,
\[ \Rightarrow {a^2} - ab + ab - {b^2}\]
Then we take the common, from the first two terms we take \[a\] as the common and from the last two we will take \[b\] as common.
Then the above expression is written as below.
\[ \Rightarrow a\left( {a - b} \right) + b\left( {a - b} \right)\]
By solving the above expression, the result would be as below.
\[\therefore \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\].