Solve the following quations by using qardratic formula:
`(2)/(3)x=-(1)/(6)x^(2)-(1)/(3)`

Answer

To solve the equation \(\frac{2}{3}x = -\frac{1}{6}x^2 - \frac{1}{3}\) using the quadratic formula, follow these steps:

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

Starting with:
\[ \frac{2}{3}x + \frac{1}{6}x^2 + \frac{1}{3} = 0 \]

To eliminate the fractions, multiply the entire equation by 6 (the least common multiple of the denominators 3 and 6):
\[ 6 \left(\frac{2}{3}x\right) + 6 \left(-\frac{1}{6}x^2\right) + 6 \left(-\frac{1}{3}\right) = 0 \]

This simplifies to:
\[ 4x - x^2 - 2 = 0 \]

Rearranging gives:
\[ -x^2 + 4x - 2 = 0 \]

Multiplying through by -1 to make the coefficient of \(x^2\) positive:
\[ x^2 - 4x + 2 = 0 \]

### Step 2: Identifying Coefficients
Now we identify the coefficients \(a\), \(b\), and \(c\):
- \(a = 1\)
- \(b = -4\)
- \(c = 2\)

### Step 3: Applying 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 \(c\):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]

### Step 4: Simplifying the Expression
Calculating \(b^2 - 4ac\):
\[ (-4)^2 = 16 \]
\[ 4 \cdot 1 \cdot 2 = 8 \]
Thus,
\[ b^2 - 4ac = 16 - 8 = 8 \]

Now substituting back into the formula:
\[ x = \frac{4 \pm \sqrt{8}}{2} \]

### Step 5: Further Simplifying
Since \(\sqrt{8} = 2\sqrt{2}\), we can write:
\[ x = \frac{4 \pm 2\sqrt{2}}{2} \]

Dividing each term by 2:
\[ x = 2 \pm \sqrt{2} \]

### Step 6: Final Roots
Thus, the roots of the equation are:
\[ x = 2 + \sqrt{2} \quad \text{and} \quad x = 2 - \sqrt{2} \]

### Summary of the Solution
The roots of the given equation are:
\[ x = 2 + \sqrt{2} \quad \text{and} \quad x = 2 - \sqrt{2} \]

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