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# Algebra

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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