Question

Using\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\], find
\[\begin{align}
  & \left( i \right){{51}^{2}}-{{49}^{2}} \\
 & \left( ii \right){{\left( 1.02 \right)}^{2}}-{{\left( 0.98 \right)}^{2}} \\
 & \left( iii \right){{153}^{2}}-{{147}^{2}} \\
 & \left( iv \right){{12.1}^{2}}-{{7.9}^{2}} \\
\end{align}\]

Answer 1

Hint: In order to solve the question, we use the formula in all the four parts, and find the values for all the options, here the \[a\]and \[b\]are replaced by the values as given in the four parts and then solved.

Complete step-by-step solution:
For solving this question, Let us first write the formula that is given in the question
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
In order to solve this, let us take all the parts one by one
i) For the first part, \[{{51}^{2}}-{{49}^{2}}\]
Here,
\[\begin{align}
  & a=51 \\
 & b=49 \\
\end{align}\]
Further solving, we get
\[\begin{align}
  &\Rightarrow {{51}^{2}}-{{49}^{2}}=\left( 51+49 \right)\left( 51-49 \right) \\
 & \Rightarrow {{51}^{2}}-{{49}^{2}}=100\times 2 \\
 & \Rightarrow {{51}^{2}}-{{49}^{2}}=200 \\
\end{align}\]

ii) For the second part, which is given as
\[{{\left( 1.02 \right)}^{2}}-{{\left( 0.98 \right)}^{2}}\]
Here,
\[\begin{align}
  & a=1.02 \\
 & b=0.98 \\
\end{align}\]
Further solving, we get the solution as
\[\begin{align}
  &\Rightarrow {{\left( 1.02 \right)}^{2}}-{{\left( 0.98 \right)}^{2}}=\left( 1.02+0.98 \right)\left( 1.02-0.98 \right) \\
 & \Rightarrow {{\left( 1.02 \right)}^{2}}-{{\left( 0.98 \right)}^{2}}=2\times 0.04 \\
 & \Rightarrow {{\left( 1.02 \right)}^{2}}-{{\left( 0.98 \right)}^{2}}=0.08 \\
\end{align}\]

iii) For third part, \[{{153}^{2}}-{{147}^{2}}\]
We get
\[\begin{align}
  & a=153 \\
 & b=147 \\
\end{align}\]
Further solving using the above formula, we get,
\[\begin{align}
  &\Rightarrow {{\left( 153 \right)}^{2}}-{{\left( 147 \right)}^{2}}=\left( 153+147 \right)\left( 153-147 \right) \\
 & \Rightarrow {{\left( 153 \right)}^{2}}-{{\left( 147 \right)}^{2}}=300\times 6 \\
 & \Rightarrow {{\left( 153 \right)}^{2}}-{{\left( 147 \right)}^{2}}=1800 \\
\end{align}\]

iv) Now for the last part, we need to find the value for
\[{{12.1}^{2}}-{{7.9}^{2}}\]
Using the formula, if we solve this part,
\[\begin{align}
  & {{12.1}^{2}}-{{7.9}^{2}}=\left( 12.1+7.9 \right)\left( 12.1-7.9 \right) \\
 & \Rightarrow {{12.1}^{2}}-{{7.9}^{2}}=20\times 4.2 \\
 & \Rightarrow {{12.1}^{2}}-{{7.9}^{2}}=84 \\
\end{align}\]
Hence, we have found the values for all the four parts as given.

Note: The formula \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\] is one of the formulae for the squares of the numbers, these formulae make it easier to calculate various values,
 For all the parts given , calculate the values by simply putting the values in the place of the variables \[a\]and \[b\], then simplify the formula and calculate the final values.