Question

If \[{x^2} = 9\], then what is the value of \[x\]?

Answer 1

Hint: We are given an expression where we are given the value of the square of the variable and we need to find the value of the variable. We can find the value by taking square roots on both sides. Also, we know, for any variable or any constant \[x\], \[\sqrt {{x^2}} = |x|\]. So, we will use these properties and find out our answer.

Complete step-by-step answer:
We are given \[{x^2} = 9\] and we need to find the value of \[x\].
We know, \[9 = 3 \times 3\] and also, \[\underbrace {a \times a \times a \times ....... \times a}_n = {a^n}\] for any \[a\] and \[n\].
So, we can write
\[ \Rightarrow 9 = \underbrace {3 \times 3}_2 = {3^2}\]
Hence, substituting \[9 = {3^2}\] in \[{x^2} = 9\], we get
\[ \Rightarrow {x^2} = 9 = {3^2}\]
Hence, we can write it as
\[ \Rightarrow {x^2} = {3^2}\]
Now, taking square root both the sides, we get
\[ \Rightarrow \sqrt {{x^2}} = \sqrt {{3^2}} \]
Now, using \[\sqrt {{a^2}} = |a|\] for any \[a\], we get
\[ \Rightarrow |x| = |3|\]
Now, \[|a| = |b| \Rightarrow a = \pm b\], we get
\[ \Rightarrow x = \pm 3\]
Hence, we get, if \[{x^2} = 9\], then \[x = \pm 3\].

Note: We usually consider that \[\sqrt {{x^2}} = \pm x\] but actually this is not true. We consider both positive and negative numbers because when we square these two numbers individually, they both result in same number i.e. \[{\left( { - a} \right)^2} = {\left( a \right)^2} = {a^2}\] but actually the correct formula to be used is \[\sqrt {{x^2}} = |x|\]. Also, we need to revise the properties of modulus function very thoroughly.