To create a quadratic equation in general form $ax^2 + bx + c = 0$ using the fractions $\frac{5}{8}$ and $\frac{2}{3}$, let's consider these as the roots of the quadratic equation.

Given roots $r_1 = \frac{5}{8}$ and $r_2 = \frac{2}{3}$, we can use the fact that a quadratic equation with roots $r_1$ and $r_2$ is given by:

$x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0$

Step 1: Find the Sum of the Roots

$r_1 + r_2 = \frac{5}{8} + \frac{2}{3}$

To add these, we need a common denominator, which is 24: $\frac{5}{8} = \frac{15}{24} \quad \text{and} \quad \frac{2}{3} = \frac{16}{24}$ $r_1 + r_2 = \frac{15}{24} + \frac{16}{24} = \frac{31}{24}$

Step 2: Find the Product of the Roots

$r_1 \cdot r_2 = \frac{5}{8} \cdot \frac{2}{3} = \frac{10}{24} = \frac{5}{12}$

Step 3: Substitute into the Quadratic Formula

Substitute $r_1 + r_2 = \frac{31}{24}$ and $r_1 \cdot r_2 = \frac{5}{12}$ into the equation:

$x^2 - \left(\frac{31}{24}\right)x + \frac{5}{12} = 0$

Step 4: Clear the Fractions

To express this equation in standard form without fractions, multiply every term by the least common multiple of the denominators (24):

$24x^2 - 31x + 10 = 0$

Final Answer

The quadratic equation in general form is: $24x^2 - 31x + 10 = 0$

Would you like further details or have any questions?

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Here are five related questions:

1. How do we derive a quadratic equation from its roots?
2. What happens if the roots of a quadratic equation are complex numbers?
3. How does the discriminant determine the nature of the roots?
4. How do you find the vertex of a quadratic equation?
5. What are other ways to form quadratic equations from given points?

Tip: Multiplying through by the least common multiple helps clear fractions and makes equations easier to handle.