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# Algebra

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Let a and b be the solutions to 5x^2 - 11x + 4 = 0.  FInd 1/a^2 + 1/b^2.

Jan 11, 2022

Note that $$\frac{1}{a^2}+\frac{1}{b^2}=\frac{a^2+b^2}{a^2b^2} =\frac{(a+b)^2 - 2ab} {(ab)^2}$$. By Vieta's, $$a+b = \frac{-(-11)}{5} = \frac{11}{5}$$ and $$ab = \frac{4}{5}$$, so $\frac{(a+b)^2 - 2ab}{(ab)^2} = \frac{\frac{121}{25} - 2 \cdot \frac{4}{5}}{\left(\frac{4}{5}\right)^2} = \frac{\frac{81}{25}}{\frac{16}{25}} = \boxed{\frac{81}{16}}.$