The problem asks us to find the solutions to the quadratic equation $4x^2 - 20x + 25 = 0$ from the given options: $10$, $\frac{3}{4}$, $\frac{1}{2}$, $5$, $-5$, $-\frac{5}{2}$, $0$, $\frac{5}{2}$, $-10$.

1. Problem Description

The problem asks us to find the solutions to the quadratic equation $4x^2 - 20x + 25 = 0$ from the given options: $10$, $\frac{3}{4}$, $\frac{1}{2}$, $5$, $-5$, $-\frac{5}{2}$, $0$, $\frac{5}{2}$, $-10$.

2. Solution Steps

We can solve the quadratic equation $4x^2 - 20x + 25 = 0$ by factoring. Notice that $4x^2 = (2x)^2$ and $25 = 5^2$. Also, $2(2x)(5) = 20x$. Therefore, the equation is a perfect square trinomial:

$(2x - 5)^2 = 0$.

Taking the square root of both sides, we have $2x - 5 = 0$.

Solving for $x$, we get $2x = 5$, so $x = \frac{5}{2}$.

Now we check which of the given options equal $\frac{5}{2}$.

The options are: $10$, $\frac{3}{4}$, $\frac{1}{2}$, $5$, $-5$, $-\frac{5}{2}$, $0$, $\frac{5}{2}$, $-10$.

Only $\frac{5}{2}$ matches the solution.

3. Final Answer

$\frac{5}{2}$