Question

If the relations between a and b are given as $\left( a+b \right)=10$ and ab=20, find ${{a}^{2}}+{{b}^{2}}\text{ and }{{\left( a-b \right)}^{2}}$

Answer 1

Hint: For finding the value of ${{a}^{2}}+{{b}^{2}}$ , use the formula ${{x}^{2}}+{{y}^{2}}={{\left( x+y \right)}^{2}}-2xy$ and substitute the values given in the question, to reach the answer. Once you get the value of ${{a}^{2}}+{{b}^{2}}$ , use the formula ${{\left( x-y \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$ to get the value of ${{\left( a-b \right)}^{2}}$ .

Complete step-by-step answer:
Let us start the solution to the above question by finding the value of ${{a}^{2}}+{{b}^{2}}$ . We know that ${{x}^{2}}+{{y}^{2}}={{\left( x+y \right)}^{2}}-2xy$ , so ${{a}^{2}}+{{b}^{2}}$ can be represented as ${{a}^{2}}+{{b}^{2}}={{\left( a+b \right)}^{2}}-2ab$ . Also, according to the given data $\left( a+b \right)=10$ and ab=20 .
$\therefore {{a}^{2}}+{{b}^{2}}={{\left( a+b \right)}^{2}}-2ab={{10}^{2}}-2\times 20=100-40=60$
Now let us move to finding the value of ${{\left( a-b \right)}^{2}}$ . We know that ${{\left( x-y \right)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$ , so we can say that ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ . Also, according to the above result and given data ${{a}^{2}}+{{b}^{2}}=60$ and ab=20. So, if use this values, we get
${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab=60-2\times 20=20$
Therefore, we can conclude that the value of ${{a}^{2}}+{{b}^{2}}\text{ and }{{\left( a-b \right)}^{2}}$ are 60 and 20, respectively, provided $\left( a+b \right)=10$ and ab=20 .

Note: While using the algebraic formulas of ${{\left( a+b \right)}^{2}}\text{ and }{{\left( a-b \right)}^{2}}$ be very careful about the signs as their results differ just by the sign of the term 2ab. Also, be very careful as the formulas we used in the above question were all related to the whole square of two terms and are not the same for the sum of the squares of two terms. It is also prescribed that you cross check the calculations of the first steps before proceeding, as an error in the first part can lead to error in finding the other derived values as well.