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# help

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If one root of (a^2 + 9)x^2 + 13x + 6a = 0 is equal to the reciprocal of the other root, then find a.

Dec 2, 2019

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If one root of (a^2 + 9)x^2 + 13x + 6a = 0 is equal to the reciprocal of the other root, then find a.

$$\begin{array}{|rcll|} \hline (a^2 + 9)x^2 + 13x + 6a &=& 0 \quad & | \quad : (a^2 + 9) \\\\ x^2 + \dfrac{13}{a^2 + 9}x + \underbrace{\dfrac{6a}{a^2 + 9}}_{=r_1r_2} &=& 0 \quad & | \quad \text{vieta} \\\\ r_1r_2 &=& \dfrac{6a}{a^2 + 9} \quad |\quad r_2 = \dfrac{1}{r_1} \\ r_1\dfrac{1}{r_1} &=& \dfrac{6a}{a^2 + 9} \\ 1 &=& \dfrac{6a}{a^2 + 9} \\ a^2 + 9 &=& 6a \\ a^2-6a + 9 &=& 0 \\\\ a &=& \dfrac{6 \pm \sqrt{6^2-4(9)} } {2} \\ a &=& \dfrac{6 \pm \sqrt{0}} {2} \\ a &=& \dfrac{6}{2} \\ \mathbf{a} &=& \mathbf{3} \\ \hline \end{array}$$

Dec 2, 2019