## Related questions with answers

Solve each open sentence.

$\frac { x ^ { 2 } } { 9 } + \frac { x - 1 } { 10 } = 0$

Solution

Verified$\begin{align*} 90 \cdot\left(\frac{x^2}{9}+\frac{x-1}{10}\right)&=90 \cdot 0 && \text{Multiply both sides by} \ 90, \\ &\text{} && \text{the LCM of} \ 9, \ \text{and} \ 10. \\ 90 \cdot \frac{x^2}{9}+90 \cdot \frac{x-1}{10}&=0 && \text{Use the Distributive Property.} \\ 10x^2+9 \cdot \left(x-1\right)&=0 && \text{Simplify.} \\ 10x^2+9x-9&=0 && \text{Use the Distributive Property.} \\ \left(5x-3\right)\left(2x+3\right)&=0 \\ \end{align*}$

The given equation already has $0$ as one side, and the other side is already factored. Simply set each factor equal to $0$ and solve:

$\begin{align*} \left(5x-3\right)&=0 \hspace{1cm} \text{or} \hspace{1cm} \left(2x+3\right)=0 \hspace{2cm} \text{Zero-Product Property} \\ 5x&=3 \hspace{1cm} \text{or} \hspace{2cm} 2x=-3 \\ x&=\frac{3}{5} \hspace{1cm} \text{or} \hspace{2.2cm} x=-\frac{3}{2} \\ \end{align*}$

The solution set is $\boxed{\color{#c34632}\left\lbrace \frac{3}{5}, -\frac{3}{2}\right\rbrace}$

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