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If a−b=7 and $a^{3} - b^{3}$ =133, then find the value of $a^{2} + b^{2}$ ?

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answer is B.

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Detailed Solution

According to the problem, we are given that a−b=7 and

a^{3} - b^{3}

=133, and we need to find the value of

a^{2} + b^{2}

.
We have a−b=7 ---(1).
Let us apply cube on both sides of the equation (1).
So, we get

{(a - b)}^{3} = 7^{3}

---(2).
We know that

{(a - b)}^{3}

a^{3} - b^{3}

−3ab(a−b) . Let us use this in equation (2).
So, we get

a^{3} - b^{3}

−3ab(a−b)=343---(3).
Let us substitute the results a−b=7 and

a^{3} - b^{3}

=133 in equation (3).
So, we get 133−3ab(7)=343 .
⇒−21ab=210
⇒ab=−10---(4).
Now, we know that

a^{3} - b^{3}

=(a−b)(

a^{2} + b^{2}

+ab) .
But we already have a−b=7,

a^{3} - b^{3}

=133 and ab=−10.
⇒133=7(

a^{2} + b^{2}

−10).
⇒19=(

a^{2} + b^{2}

−10).
⇒

a^{2} + b^{2}

=29.
∴We have found the value of

a^{2} + b^{2}

as 29.

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