Solve the following quations by using qardratic formula:
`(1)/(15)x^(2)+(5)/(3)=(2)/(3)x`

Answer

To solve the quadratic equation \(\frac{1}{15}x^2 + \frac{5}{3} = \frac{2}{3}x\) using the quadratic formula, we will follow these steps:

### Step 1: Rearrange the equation
First, we need to rearrange the equation into the standard form \(ax^2 + bx + c = 0\).

Starting with:
\[ \frac{1}{15}x^2 + \frac{5}{3} - \frac{2}{3}x = 0 \]

### Step 2: Eliminate fractions
To eliminate the fractions, we can multiply the entire equation by 15 (the least common multiple of the denominators):

\[ 15 \left(\frac{1}{15}x^2\right) + 15 \left(\frac{5}{3}\right) - 15 \left(\frac{2}{3}x\right) = 0 \]

This simplifies to:
\[ x^2 + 25 - 10x = 0 \]

### Step 3: Rearrange into standard form
Now, we rearrange it to the standard form:
\[ x^2 - 10x + 25 = 0 \]

### Step 4: Identify coefficients
From the standard form \(ax^2 + bx + c = 0\), we identify:
- \(a = 1\)
- \(b = -10\)
- \(c = 25\)

### Step 5: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \]

### Step 6: Simplify the expression
Calculating the discriminant:
\[ x = \frac{10 \pm \sqrt{100 - 100}}{2} \]
\[ x = \frac{10 \pm \sqrt{0}}{2} \]
\[ x = \frac{10 \pm 0}{2} \]
\[ x = \frac{10}{2} \]
\[ x = 5 \]

### Final Solution
The solution to the equation is:
\[ x = 5 \]