asqrt(a)+bsqrt(b)=183 and asqrt(b)+bsqrt(a)=182 Find 9/5(a+b) ?

2 Answers
Apr 5, 2018

73

Explanation:

Let sqrta =x => x^2=a
and sqrtb =y => y^2=b

So, the altered equations now look like,
x^ 3 +y^ 3 =183 and x^2y+y^ 2x=182
color(white)(wwwwwwwwwd =>xy(x+y) = 182

We know the basic identity,
color(magenta)((x+y)^3=x^ 3 +y^ 3 +3(x^ 2 y+y^ 2 x)

Plugging in the "altered" equations,
=>(x+y)^3=183 +3(182)
=>(x+y)^3=729
=> color(blue)(x+y = 9

Now, plug x+y = 9 into xy(x+y) = 182
=>color(red)( xy = 182/9

Also, (x-y)^2 = (color(blue)(x+y))^2 - 4color(red)(xy)

=> color(blue)(9)^2 - 4(color(red)(182/9))

=> 81-(4(182))/9 = 1/9

=> x-y = +-1/3

Since we've found out x+y = 9 and x-y = +-1/3, we can find the values of x and y by elimination method.

When x-y = + 1/3 color(white)(wwwwwwwwwd When x-y = - 1/3

Adding both the equations, color(white)(wwwwwd Adding both the equations,

2x = 9+1/3 color(white)(wwwwwwwwwwwwwwd 2x = 9-1/3

x= 14/3 color(white)(wwwwwwwwwwwwwwwwwwx= 13/3

therefore, color(white)(wwwwwwwwwwwwwwwwd therefore,

y=13/3 color(white)(wwwwwwwwwwwwwwwwwd y=14/3

Either way, we have to find, 9/5 (a+b) => 9/5 (x^2 + y^2)

=> 9/5 ((13/3)^2 + (14/3)^2)

=> cancel9/5 xx (169 + 196) /cancel9

=>73

Apr 5, 2018

Alternate to the previous answer

Explanation:

Let sqrta =x => x^2=a
and sqrtb =y => y^2=b

So, the altered equations now look like,
x^ 3 +y^ 3 =183 and x^2y+y^ 2x=182
color(white)(wwwwwwwwwd =>xy(x+y) = 182

We know the basic identity,
color(magenta)((x+y)^3=x^ 3 +y^ 3 +3(x^ 2 y+y^ 2 x)

Plugging in the "altered" equations,
=>(x+y)^3=183 +3(182)
=>(x+y)^3=729
=> color(blue)(x+y = 9

Now, plug x+y = 9 into xy(x+y) = 182
=>color(red)( xy = 182/9

We know, (x+y)^2 = x^2 + y^2 +2xy
=> x^2 + y^2 = (x+y)^2 - 2xy
color(white)(wwwwwwi = 9^2 - 2(182/9)

color(white)(wwwwwwi = 365/9

We need to find, 9/5 (a+b) => 9/5 (x^2 + y^2)

=> 9/5 (365/9)

=> cancel9/5 xx (365) /cancel9

=>73

color(white)(wwwwwwi

color(white)(wwwwwwi

color(white)(wwwwwwi

Another way could be,
since,
x³+y³=183

(x+y)(x²+y²−xy)=183

(x+y)(x²+y²)−xy(x+y)=183

(x+y)(x²+y²)=182+183 =365

(x²+y²)=365/9
.
Then the required expression = (9/5)(a+b)=(9/5)(x²+y²)
=(9/5)(3659)=73