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If a + b = 7 and a^3 + b^3 = 42, what is the value of the sum a^2 + b^2? Express your answer as a common fraction.

 May 14, 2022
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\(a+b=7\)

\((a+b)^2=7^2\)

\(a^2+2ab+b^2=49\)

 

\((a+b)^3=7^3\)

\(a^3 + 3a^2b+3ab^2+b^3=343\)

\(3a^2b+3ab^2=301\)

\(a^2b+ab^2=\frac{301}{3}\)

\(ab(a+b)=\frac{301}{3}\)

\(ab(7)=\frac{301}{3}\)

\(ab = \frac{43}{3}\)

 

\(a^2 + 2ab + b^2 = 49\)

\(a^2 + \frac{86}{3} + b^2 = 49\)

\(a^2 + b^2 = 49 - \frac{86}{3} \)

\(a^2 + b^2 = \boxed{\frac{61}{3}}\)

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 May 14, 2022

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