Question

Find the square of the following number:
205

Answer 1

Hint:In this question, to find the square of the given number we need to express it as the sum of two numbers whose square is easy to find and then expand it accordingly using the factorisation of the polynomials formula. Then simplify it further to get the result.
\[{{\left( x+a \right)}^{2}}={{x}^{2}}+{{a}^{2}}+2ax\]

Complete step-by-step answer:
Now, the number given in the question is 205
POLYNOMIAL:An expression of the form \[{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+{{a}_{2}}{{x}^{n-2}}+....+{{a}_{n-1}}x+{{a}_{n}}\], where a0, a1,....., an are real numbers and n is a non-negative integer, is called a polynomial.
As we already know from the factorisation of polynomials that
\[{{\left( x+a \right)}^{2}}={{x}^{2}}+2ax+{{a}^{2}}\]
Now, the square of the given number can be written as
\[\Rightarrow {{205}^{2}}\]
Now, this can be further written as the sum of two numbers whose square is eay to express
\[\Rightarrow {{\left( 200+5 \right)}^{2}}\]
Now, let us expand this using the above factorisation of polynomials formula
Now, on comparing this with the above formula we get,
\[x=200,a=5\]
Now, let us substitute these values in the formula to simplify it further
\[\Rightarrow {{200}^{2}}+{{5}^{2}}+2\times 200\times 5\]
Now, this can be further written in the simplified form as
\[\Rightarrow 40000+25+2000\]
Now, on further simplification we get,
\[\Rightarrow 42025\]
Hence, the square of 205 is 42025.

Note:Instead of writing 205 as a sum of two numbers we can also solve it by writing it as a sum of three small numbers and then simplify it using the respective factorisation of polynomials formula and further simplify it. Both the methods give the same result.It is important to note that though we can directly find the square by multiplying the number twice this method would be preferable because multiplying a big number twice is a little bit confusing.