$$p+\frac{10}{p}-7 = \frac{8}{9}$$
Answer
$$ \begin{matrix}p_1 = \dfrac{ 71 }{ 18 }-\dfrac{\sqrt{ 1801 }}{ 18 } & p_2 = \dfrac{ 71 }{ 18 }+\dfrac{\sqrt{ 1801 }}{ 18 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} p+\frac{10}{p}-7 &= \frac{8}{9}&& \text{multiply ALL terms by } \color{blue}{ p\cdot9 }. \\[1 em]p\cdot9p+p\cdot9\cdot\frac{10}{p}-p\cdot9\cdot7 &= p\cdot9\cdot\frac{8}{9}&& \text{cancel out the denominators} \\[1 em]9p^2+90-63p &= 8p&& \text{simplify left side} \\[1 em]9p^2-63p+90 &= 8p&& \text{move all terms to the left hand side } \\[1 em]9p^2-63p+90-8p &= 0&& \text{simplify left side} \\[1 em]9p^2-71p+90 &= 0&& \\[1 em] \end{aligned} $$
$ 9x^{2}-71x+90 = 0 $ is a quadratic equation.
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