Question
$$\frac{1}{2}x\cdot\frac{3}{2}(x+1)-\frac{1}{4} = 5$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 29 }}{ 2 } & x_2 = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 29 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{2}x\cdot\frac{3}{2}(x+1)-\frac{1}{4} &= 5&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1}{2}x\cdot\frac{3}{2}(x+1)-4\cdot\frac{1}{4} &= 4\cdot5&& \text{cancel out the denominators} \\[1 em]3x^2+3x-1 &= 20&& \text{move all terms to the left hand side } \\[1 em]3x^2+3x-1-20 &= 0&& \text{simplify left side} \\[1 em]3x^2+3x-21 &= 0&& \\[1 em] \end{aligned} $$
$ 3x^{2}+3x-21 = 0 $ is a quadratic equation.
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