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: Eliminate the fractions
Multiply the entire equation by 15 to eliminate the denominators:

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

This simplifies to:

\[ x^2 + 25 = 10x \]

### Step 2: Rearrange the equation
Rearranging the equation to standard quadratic form \(ax^2 + bx + c = 0\):

\[ x^2 - 10x + 25 = 0 \]

### Step 3: Identify coefficients
From the equation \(x^2 - 10x + 25 = 0\), we can identify the coefficients:

- \(a = 1\)
- \(b = -10\)
- \(c = 25\)

### Step 4: 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 \(c\):

\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \]

This simplifies to:

\[ x = \frac{10 \pm \sqrt{100 - 100}}{2} \]

### Step 5: Simplify the expression
Calculating the discriminant:

\[ x = \frac{10 \pm \sqrt{0}}{2} \]

Since \(\sqrt{0} = 0\), we have:

\[ x = \frac{10}{2} \]

### Step 6: Final solution
Thus, the solution is:

\[ x = 5 \]