Solving a Quadratic Equation
The equation in the image is:
\[ 2r + 2 \sqrt{r^2} + \frac{r}{2} = 10 \]
To solve this equation for \( r \), follow these steps:
1. Simplify the square root term. Since \( \sqrt{r^2} \) is the square root of a square, it simplifies to \( r \) if \( r \geq 0 \) (assuming \( r \) is non-negative, as you cannot take the square root of a negative number in real numbers).
So, you have:
\[ 2r + 2r + \frac{r}{2} = 10 \]
\[ 4r + \frac{r}{2} = 10 \]
2. Combine like terms by finding a common denominator for the \( r \)-terms.
Multiply each term by 2 to get rid of the fraction:
\[ 2 \cdot 4r + 2 \cdot \frac{r}{2} = 2 \cdot 10 \]
\[ 8r + r = 20 \]
\[ 9r = 20 \]
3. Divide both sides by 9 to solve for \( r \):
\[ r = \frac{20}{9} \]
Therefore, \( r = \frac{20}{9} \) or approximately 2.2222.
