Question

If a^2+b^2=117,ab=54,find the value of (a+b), (a-b) and a^2-b^2

Answer 1

first take 1st eq and solve for value of X and substitute for y

Answer 2

$\huge\bf\mathscr\pink{Your\: Answer}$

(a+b) = 15

(a-b) = 3

${a}^{2}$ = 81

${b}^{2}$ = 36

step-by-step explanation:

Given,

${a}^{2}$ +${b}^{2}$ = 117

.................(i)

adding 2ab on both sides,

we get,

${a}^{2}$ ${b}^{2}$ + 2ab= 117 +2ab

..............(ii)

But,

it is given that,

ab = 54

=> 2ab = 2×54 = 108

so,

putting the value of 2ab in eqn (ii),

we get,

${(a+b)}^{2}$ = 117 + 108

=> ${(a+b)}^{2}$ = 225

=> a+b = √225

=> a+ b = 15 .............(iii)

Now,

Subtracting 2ab from both sides in eqn (i)

we get,

${(a-b)}^{2}$ = 117- 108

=> ${(a-b)}^{2}$ = 9

=> a- b = √9

=> a-b = 3 ................(iv)

Now,

adding eqn (iii) and (iv),

we get,

2a = 18

=> a = 18/2 = 9

Putting the value of a in eqn (iii),

we get,

9 + b = 15

=> b = 15- 9 = 6

so,

${a}^{2}$ = ${9}^{2}$ = 81

and,

${b}^{2}$= ${6}^{2}$ = 36