To solve the quadratic equation \( 4\left(\frac{x}{5}\right)^2 - \frac{16}{5}x + 15 = 0 \), we can follow these steps:

### Step 1: Simplify the equation
Start by eliminating the fractions by multiplying the entire equation by 25 (the least common multiple of the denominators).

\[
25 \times \left(4\left(\frac{x}{5}\right)^2 - \frac{16}{5}x + 15\right) = 25 \times 0
\]

This simplifies to:

\[
4x^2 - 80x + 375 = 0
\]

### Step 2: Solve using the quadratic formula
The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 4x^2 - 80x + 375 = 0 \), the coefficients are:
- \( a = 4 \)
- \( b = -80 \)
- \( c = 375 \)

Substitute these into the quadratic formula:

\[
x = \frac{-(-80) \pm \sqrt{(-80)^2 - 4(4)(375)}}{2(4)}
\]

\[
x = \frac{80 \pm \sqrt{6400 - 6000}}{8}
\]

\[
x = \frac{80 \pm \sqrt{400}}{8}
\]

\[
x = \frac{80 \pm 20}{8}
\]

### Step 3: Calculate the solutions
Now, compute the two possible values for \( x \):

1. \( x = \frac{80 + 20}{8} = \frac{100}{8} = 12.5 \)
2. \( x = \frac{80 - 20}{8} = \frac{60}{8} = 7.5 \)

### Final Answer:
The solutions to the equation are \( x = 12.5 \) and \( x = 7.5 \).

Would you like further details or have any questions?

Here are some related questions:
1. How do you factor quadratic equations?
2. What is the discriminant in the quadratic formula?
3. Can the quadratic formula be derived?
4. What does each part of the quadratic formula represent?
5. How do you graph quadratic equations?

**Tip:** Always check your solutions by substituting them back into the original equation.

Learn how to solve the quadratic equation 4((x/5)^2) - (16/5)x + 15 = 0 with a step-by-step solution. This problem involves manipulating fractions and applying the quadratic formula. Suitable for high school students (Grades 9-12).

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