To rewrite the expression \( 7x - x^2 \) by completing the square, let's go through the steps.

### Step 1: Write the quadratic expression in standard form.
The standard form for a quadratic expression is \( ax^2 + bx + c \). Rewriting \( 7x - x^2 \) in standard form:
\[
y = -x^2 + 7x
\]
This can be rewritten as:
\[
y = -(x^2 - 7x)
\]

### Step 2: Complete the square inside the parentheses.
To complete the square, take half of the coefficient of the linear term \(-7x\), square it, and add it inside the parentheses. The coefficient of \(x\) is \(-7\), so:
\[
\left(\frac{-7}{2}\right)^2 = \left(\frac{-7}{2}\right)^2 = \frac{49}{4}
\]
Now add and subtract \(\frac{49}{4}\) inside the parentheses:
\[
y = -\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4}\right)
\]
\[
y = -\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4}\right)
\]

### Step 3: Distribute the negative sign and simplify.
Distribute the negative sign across the terms inside the parentheses:
\[
y = -\left(x - \frac{7}{2}\right)^2 + \frac{49}{4}
\]

### Result:
The expression \( 7x - x^2 \) can be rewritten by completing the square as:
\[
y = -\left(x - \frac{7}{2}\right)^2 + \frac{49}{4}
\]

This form shows that the parabola opens downwards (because of the negative sign) and has its vertex at \(\left(\frac{7}{2}, \frac{49}{4}\right)\).

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Would you like more details on any of these steps or have any questions?

### Related Questions:
1. How does completing the square change when the leading coefficient is negative?
2. How do you find the vertex of a quadratic equation using the vertex form?
3. Can completing the square help in solving quadratic equations with fractional coefficients?
4. What is the significance of the vertex in the graph of a quadratic function?
5. How can you determine the direction in which a parabola opens?

### Tip:
When completing the square, always pay attention to the sign of the leading coefficient. If it's negative, factor it out before completing the square to avoid mistakes in your calculations.

Learn how to complete the square to solve the expression 7x - x^2 with detailed steps. This problem involves algebraic concepts like completing the square and quadratic equations, suitable for Grades 10-12 students.

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