To write a quadratic equation given its solutions (also called roots), you can use the factored form of the quadratic equation:

$a(x - r_1)(x - r_2) = 0$

Where $r_1$ and $r_2$ are the roots of the equation.

In this case, the given solutions are $\frac{3}{2}$ and $\frac{5}{8}$. Therefore, we can write the equation as:

$a(x - \frac{3}{2})(x - \frac{5}{8}) = 0$

We can expand this equation step by step:

Step 1: Multiply the two binomials

First, expand $(x - \frac{3}{2})(x - \frac{5}{8})$:

$(x - \frac{3}{2})(x - \frac{5}{8}) = x^2 - \left(\frac{3}{2} + \frac{5}{8}\right)x + \left(\frac{3}{2} \cdot \frac{5}{8}\right)$

Now, simplify the coefficients:

1. $\frac{3}{2} + \frac{5}{8} = \frac{12}{8} + \frac{5}{8} = \frac{17}{8}$
2. $\frac{3}{2} \cdot \frac{5}{8} = \frac{15}{16}$

So the expanded expression is:

$x^2 - \frac{17}{8}x + \frac{15}{16}$

Step 2: Clear the fractions

To eliminate the fractions, multiply the entire equation by 16 (the least common denominator):

$16(x^2 - \frac{17}{8}x + \frac{15}{16}) = 0$

Distribute the 16:

$16x^2 - 34x + 15 = 0$

Final Answer:

The quadratic equation in standard form is:

$16x^2 - 34x + 15 = 0$

Would you like further details on any of the steps? Let me know if you need additional explanations.

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Related questions:

1. How do you find the solutions of a quadratic equation using the quadratic formula?
2. How do you factor a quadratic equation like $x^2 - 7x + 10 = 0$?
3. How do you convert a quadratic equation from factored form to standard form?
4. What are the conditions for a quadratic equation to have real or complex solutions?
5. What is the discriminant, and how does it help in identifying the nature of the roots of a quadratic equation?

Tip: Always check the solutions of the quadratic by substituting them back into the original equation to confirm they satisfy it!