Question

Solve it:
If a+b = 10 and ab =16, then find the value of a^2+ab+b^2.

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Answer 1

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Answer 2

$\huge\bf\green {ANSWER}$


In the given problem, we have to find the value of ( a^2 + ab + b^2 )


Given : a + b = 10 and ab = 16


We shall use the identity ( a + b )^3 = a^3 + b^3 + 3ab( a + b )


We can arrange this identity as


a^3 + b^3 = ( a + b )^3 - 3ab( a + b )


a^3 + b^3 = ( 10 )^3 - 3 × 16 (10)


a^3 + b^3 = 1000 - 480


a^3 + b^3 = 520


Now Substituting value in a^3 + b^3 = ( a + b ) ( a^2 + b^2 - ab ) as


a^3 + b^3 = 520 , a + b = 10


a^3 + b^3 = ( a + b ) ( a^2 + b^2 - ab )


520 = ( 10 )( a^2 + b^2 - ab )


520 / 10 = ( a^2 + b^2 - ab )


52 = ( a^2 + b^2 - ab )


We can write a^2 + b^2 + ab as a^2 + b^2 + ab - 2ab + 2ab


Now rearrange a^2 + b^2 + ab - 2ab + 2ab as


= a^2 + b^2 + ab - 2ab + 2ab


= ( a + b )^2 - ab


Thus a^2 + b^2 + ab = ( a + b )^2 - ab


Now Substituting values ( a + b ) = 10 and ab = 16


a^2 + b^2 + ab = ( 10 )^2 - 16


a^2 + b^2 + ab = 100 - 16


a^2 + b^2 + ab = 84


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