Question

If $ a+b=5 $ and $ ab=6 $ , then find $ {{a}^{2}}+{{b}^{2}} $ .

Answer 1

Hint: We will first take squares on both sides of equation $ a+b=5 $ . Then, we will use the formula of $ {{\left( a+b \right)}^{2}} $ which is given as $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ . Then, on solving we will keep all other terms except $ {{a}^{2}}+{{b}^{2}} $ on right hand side and on substituting values given to use, we will get our answer.

Complete step-by-step answer:
Here, we are given with the data $ a+b=5 $ and $ ab=6 $ . So, we will first take squares on both side of the equation $ a+b=5 $ .
So, we get as
 $ {{\left( a+b \right)}^{2}}={{\left( 5 \right)}^{2}} $
Now, we will use the formula i.e. $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ . So, we can write it as
 $ {{a}^{2}}+{{b}^{2}}+2ab=25 $
Here, we are asked to find the value of $ {{a}^{2}}+{{b}^{2}} $ . So, we will keep that term on the left hand side and rest all terms on the right hand side.
We get as
 $ {{a}^{2}}+{{b}^{2}}=25-2ab $
Now, we are given with value $ ab=6 $ . On substituting this, in above equation we get as
 $ {{a}^{2}}+{{b}^{2}}=25-2\left( 6 \right)=25-12 $
 $ {{a}^{2}}+{{b}^{2}}=13 $
Thus, we got the value of $ {{a}^{2}}+{{b}^{2}} $ equals 13.

Note: Another way to solve this by taking the equation $ {{a}^{2}}+{{b}^{2}} $ and making it perfect square by adding and subtracting $ 2ab $ . So, we get as $ {{a}^{2}}+{{b}^{2}}+2ab-2ab $ . So, on further solving we can write it as $ {{\left( a+b \right)}^{2}}-2ab $ . So, on putting the values given to us we will get the answer. We get as
 $ \Rightarrow {{\left( 5 \right)}^{2}}-2\cdot 6=25-12=13 $ . Thus, the same answer we will get.