To complete the square for the quadratic expression $-4.9x^2 + 15x$, we need to rewrite it in the form:

$a(x - h)^2 + k$

where $a$ is the coefficient of $x^2$, and $(h, k)$ is the vertex of the quadratic function.

Step-by-Step Process to Complete the Square

1. Factor out the coefficient of $x^2$ (if it is not 1):

   The coefficient of $x^2$ is $-4.9$. We factor this out from the terms involving $x$:

   $-4.9(x^2 - \frac{15}{4.9}x)$

2. Determine the term needed to complete the square:

   To complete the square, we take half of the coefficient of $x$ (which is $-\frac{15}{4.9}$), square it, and add it inside the parenthesis. The coefficient of $x$ is:

   $-\frac{15}{4.9} \approx -3.0612$

   Half of this is:

   $-\frac{15}{2 \times 4.9} = -\frac{15}{9.8} \approx -1.5306$

   Squaring this value gives:

   $\left(-\frac{15}{9.8}\right)^2 = \frac{225}{96.04} \approx 2.3427$

3. Add and subtract this term inside the parenthesis:

   We add and subtract this term inside the parentheses to complete the square:

   $-4.9 \left( x^2 - \frac{15}{4.9}x + \frac{225}{96.04} - \frac{225}{96.04} \right)$

   This simplifies to:

   $-4.9 \left( \left(x - \frac{15}{9.8}\right)^2 - \frac{225}{96.04} \right)$

4. Distribute and simplify:

   Now, distribute the $-4.9$:

   $-4.9 \left(x - \frac{15}{9.8}\right)^2 + 4.9 \cdot \frac{225}{96.04}$

   Simplify the constant term:

   $-4.9 \left(x - \frac{15}{9.8}\right)^2 + \frac{1102.5}{96.04} \approx -4.9 \left(x - 1.53\right)^2 + 11.48$

Completed Square Form:

The expression $-4.9x^2 + 15x$ in completed square form is:

$-4.9 \left(x - 1.53\right)^2 + 11.48$

This shows that the vertex of the parabola is at $(1.53, 11.48)$.

Would you like more details on any step or explore further?

Related Questions:

1. How do you find the vertex of a quadratic function using the completed square form?
2. What does the completed square form tell us about the graph's direction and vertex?
3. How would you use the quadratic formula to find the roots of this function?
4. Can you use the vertex form to determine the maximum or minimum value of this function?
5. What are some real-world applications of completing the square?

Tip: Completing the square is useful for converting a quadratic function into vertex form, which makes it easy to identify the vertex and graph the function.