Question

If \[9\sqrt{x}=\sqrt{12}+\sqrt{147}\] , then the value of \[x\] is :
A. \[-2\]
B. \[2\]
C. \[3\]
D. \[-3\]

Answer 1

Hint: Firstly we will write the given equation and then we will find out the given factors of \[12\] and \[147\] then we will write the factors of the equation. Then we take factors in common, after that add the terms and find the value of \[x\] and check which option is correct in the given options.

Complete step by step answer:
Factor, in mathematics a number or algebraic expression that divides another number or expression evenly that is with no remainder.
There are no shortcuts to getting better at identifying factors and multiples. Factoring a polynomial means writing it as a product of other polynomials.
In algebraic expressions, terms are formed as products of factors. For example, in the algebraic expression \[5xy+3x\] the term \[5xy\] has been formed by the factors \[5,x\] and \[y\].
Observe that the \[5,x\] and \[y\] of \[5xy\] cannot further be expressed as a product of factors. We may say that \[5,x\] and \[y\] are prime factors of \[5xy\] . In algebraic expressions, we use the word irreducible in the place of prime. We say that \[5\times x\times y\] is the irreducible form of \[5xy\]
We must note that \[5\times (xy)\] is not an irreducible form of \[5xy\] , since the factor \[xy\] can be further expressed as a product of \[x\] and \[y\] i.e, \[xy=x\times y\]
When we factorize an algebraic expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
Expressions like \[3xy,5{{x}^{2}}y,2x(y+2),5(y+1)(x+2)\] are already in factor form. Their factors can be just read off from them, as we already know.
On the other hand expressions like \[2x+4,3x+3y,{{x}^{2}}+5x,{{x}^{2}}+5x+6\] . It is not obvious what their factors are. We need to develop a systematic method to factorize these expressions, i.e, to find their factors.
Now according to the question:
We have given that: \[9\sqrt{x}=\sqrt{12}+\sqrt{147}\]
Now we will write the factors of \[12\] and \[147\]
The factor of \[12\] can be written as \[2\times 2\times 3\]
The factor of \[147\] can be written as \[3\times 7\times 7\]
\[\Rightarrow \]\[9\sqrt{x}=\sqrt{3\times 2\times 2}+\sqrt{3\times 7\times 7}\]
\[\Rightarrow \]\[9\sqrt{x}=2\sqrt{3}+7\sqrt{3}\]
Adding both the terms we will get:
\[\Rightarrow \]\[9\sqrt{x}=9\sqrt{3}\]
\[\Rightarrow \]\[\sqrt{x}=\sqrt{3}\]
Hence, \[x=3\]

So, the correct answer is “Option C”.

Note: We must remember that factors are always whole numbers or integers and never decimal or fractions and all the even numbers will have \[2\] in their factors. The factor of a number is always less than or equal to the given number. Division and multiplication are the operations that are used in finding the factors.