To find the vertex of the parabola given by the equation \( y = x^2 + 5x \), we can use the formula for the vertex of a parabola in the standard form \( y = ax^2 + bx + c \).
The x-coordinate of the vertex can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \( a = 1 \) and \( b = 5 \). Plugging these values into the formula gives:
\[ x = -\frac{5}{2 \cdot 1} = -\frac{5}{2} \]
Next, we need to find the y-coordinate of the vertex by substituting \( x = -\frac{5}{2} \) back into the original equation:
\[ y = \left(-\frac{5}{2}\right)^2 + 5\left(-\frac{5}{2}\right) \]
Calculating \( \left(-\frac{5}{2}\right)^2 \):
\[ \left(-\frac{5}{2}\right)^2 = \frac{25}{4} \]
Calculating \( 5\left(-\frac{5}{2}\right) \):
\[ 5\left(-\frac{5}{2}\right) = -\frac{25}{2} \]
Now, we need to combine these two results:
\[ y = \frac{25}{4} - \frac{25}{2} \]
To combine the fractions, we need a common denominator. The denominator of \( \frac{25}{2} \) can be converted to 4:
\[ -\frac{25}{2} = -\frac{25 \cdot 2}{2 \cdot 2} = -\frac{50}{4} \]
Thus, we can rewrite \( y \):
\[ y = \frac{25}{4} - \frac{50}{4} = \frac{25 - 50}{4} = \frac{-25}{4} \]
Therefore, the vertex of the parabola \( y = x^2 + 5x \) is at:
\[ \left(-\frac{5}{2}, -\frac{25}{4}\right) \]