Find the equation whose root are 5/6 and -3/4

Let the roots be $r_1 = \frac{5}{6}$ and $r_2 = -\frac{3}{4}$.

$$\text{Sum of roots} = r_1 + r_2 = \frac{5}{6} + \left(-\frac{3}{4}\right)$$

To add these fractions, find a common denominator, which is 12.

$$\text{Sum of roots} = \frac{5 \times 2}{6 \times 2} - \frac{3 \times 3}{4 \times 3} = \frac{10}{12} - \frac{9}{12} = \frac{1}{12}$$

$$\text{Product of roots} = r_1 \times r_2 = \frac{5}{6} \times \left(-\frac{3}{4}\right)$$ $$\text{Product of roots} = -\frac{5 \times 3}{6 \times 4} = -\frac{15}{24}$$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 3.

$$\text{Product of roots} = -\frac{15 \div 3}{24 \div 3} = -\frac{5}{8}$$

The general form of a quadratic equation with roots $r_1$ and $r_2$ is $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.
Substitute the sum and product calculated in the previous steps:

$$x^2 - \left(\frac{1}{12}\right)x + \left(-\frac{5}{8}\right) = 0$$ $$x^2 - \frac{1}{12}x - \frac{5}{8} = 0$$

The least common multiple (LCM) of the denominators 12 and 8 is 24. Multiply the entire equation by 24:

$$24 \left(x^2 - \frac{1}{12}x - \frac{5}{8}\right) = 24 \times 0$$ $$24x^2 - 24 \times \frac{1}{12}x - 24 \times \frac{5}{8} = 0$$ $$24x^2 - 2x - (3 \times 5) = 0$$ $$24x^2 - 2x - 15 = 0$$

The equation whose roots are $\frac{5}{6}$ and $-\frac{3}{4}$ is: