Question

How do you solve \[{x^2} = 144\]?

Answer 1

Hint: Here, will do the prime factorization of the given number and express it as a product of prime factors. Then, we will consider a pair of the same numbers as a single number because we are required to find the square root. Hence, multiplying the remaining factors, we will find the required square root of the given number, which will be the required value of \[x\].

Complete step-by-step answer:
We are given, \[{x^2} = 144\]
Now taking square root on both sides, we get,
\[x = \pm \sqrt {144} \]……………………………………….\[\left( 1 \right)\]
In order to find the square root of 144, we will do the prime factorization of 144.
Now, factorization is a method of writing an original number as the product of its various factors.
Also, prime numbers are those numbers that are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.
Hence, 144 can be written as:
\[144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\]
Now, since, we are required to find the square root,
We will take only one prime number out of a pair of the same prime numbers.
\[ \Rightarrow \sqrt {144} = 2 \times 2 \times 3\]
Multiplying the terms, we get
\[ \Rightarrow \sqrt {144} = 12\]
Substituting this value in equation \[\left( 1 \right)\], we get
 \[x = \pm \sqrt {144} = \pm 12\]

Hence, the required value of \[x\] is \[ \pm 12\]
Thus, this is the required answer.

Note:
We can solve this question using an alternate method. We will first write the given equation as the standard form of a quadratic equation.
The standard form of the quadratic equation is given by \[a{x^2} + bx + c = 0\].
So, we can write the given equation as:
\[{x^2} - 144 = 0\]
Comparing this equation with the standard form, we get
\[a = 1,b = 0\] and \[c = - 144\]
Now substituting these values in quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we get
\[x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4 \times 1 \times - 144} }}{2}\]
Simplifying the equation, we get
\[ \Rightarrow x = \dfrac{{ \pm \sqrt {{{\left( 2 \right)}^2} \times 144} }}{2}\]
Rewriting above equation, we get
\[ \Rightarrow x = \dfrac{{ \pm 2\sqrt {144} }}{2}\]
Cancelling the similar terms, we get
\[ \Rightarrow x = \pm \sqrt {144} \]
We know that square root of 144 is 12, so
\[ \Rightarrow x = \pm 12\]
Hence, this is the required answer.