Question

If a – b = 1 and ab = 12 then the value of ${{a}^{2}}+{{b}^{2}}$ is equal to:
A) 26
B) 28
C) 29
D) 25

Answer 1

Hint: It is given that a – b = 1 then do the square on both the sides and substitute the value of ab given in question and then solve. After rearranging the expressions and values on both the sides you will be left with the value of this expression ${{a}^{2}}+{{b}^{2}}$.
Complete step-by-step answer:
It is given that a – b = 1.
Squaring on both the sides of the above equation we get,
${{\left( a-b \right)}^{2}}=1$
We know that ${{\left( a-b \right)}^{2}}$ is an identity which is equal to ${{a}^{2}}+{{b}^{2}}-2ab$ so substituting this identity in the above equation we get,
${{a}^{2}}+{{b}^{2}}-2ab=1$
The value of ab is given in the question which is equal to 12. Substituting the value of ab in above equation we get,
$\begin{align}
  & {{a}^{2}}+{{b}^{2}}-2(12)=1 \\
 & \Rightarrow {{a}^{2}}+{{b}^{2}}=1+24 \\
 & \Rightarrow {{a}^{2}}+{{b}^{2}}=25 \\
\end{align}$
From the above calculation, the value of ${{a}^{2}}+{{b}^{2}}$ is 25.
Hence, the correct option is (D).

Note: The possible blunders that could happen in the question are as follows:
 We could have written${{\left( a-b \right)}^{2}}\text{ as }{{a}^{2}}+{{b}^{2}}+2ab$ and ${{\left( a-b \right)}^{2}}\text{ as }{{a}^{2}}-{{b}^{2}}$ and then would have thought where this ab has come from.
So, be careful about the signs of the identity and the expression of identity. In the hastiness of solving questions in exams these problems could be possible.