To determine where the banner should be attached to the archway, we need to find the points where the two equations intersect. The equations represent two curves: the archway and the banner.

We are given the two equations:

1. The archway: $y = -x^2 + 6x$
2. The banner: $4y = 21 - x$ (which can be rewritten as $y = \frac{21 - x}{4}$)

To find the points of intersection, we substitute the expression for $y$ from the second equation into the first equation.

Step 1: Substitute $y = \frac{21 - x}{4}$ into $y = -x^2 + 6x$

$\frac{21 - x}{4} = -x^2 + 6x$

Step 2: Eliminate the fraction by multiplying both sides of the equation by 4

$21 - x = 4(-x^2 + 6x)$ $21 - x = -4x^2 + 24x$

Step 3: Rearrange the terms into a standard quadratic form

$4x^2 - 25x + 21 = 0$

Step 4: Solve the quadratic equation using the quadratic formula

The quadratic formula is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $4x^2 - 25x + 21 = 0$, the coefficients are:

• $a = 4$
• $b = -25$
• $c = 21$

Now, applying the quadratic formula: $x = \frac{-(-25) \pm \sqrt{(-25)^2 - 4(4)(21)}}{2(4)}$ $x = \frac{25 \pm \sqrt{625 - 336}}{8}$ $x = \frac{25 \pm \sqrt{289}}{8}$ $x = \frac{25 \pm 17}{8}$

Step 5: Find the two possible values for $x$

$x_1 = \frac{25 + 17}{8} = \frac{42}{8} = 5.25$ $x_2 = \frac{25 - 17}{8} = \frac{8}{8} = 1$

Step 6: Find the corresponding $y$-values

Now substitute these $x$-values back into either equation to find the corresponding $y$-values. Let's use the equation $y = \frac{21 - x}{4}$.

For $x_1 = 5.25$: $y = \frac{21 - 5.25}{4} = \frac{15.75}{4} = 3.9375$

For $x_2 = 1$: $y = \frac{21 - 1}{4} = \frac{20}{4} = 5$

Step 7: Conclusion

The points where the banner should be attached are approximately:

• $(1, 5)$
• $(5.25, 3.94)$

Thus, the correct answer is D. (1, 5) and (5.25, 3.94).

Would you like more details on any step or further clarification on how the quadratic equation was solved?

Additional Questions:

1. How does the quadratic formula work for solving equations?
2. What are the different ways to graph a quadratic equation?
3. How do you check if your solution to a system of equations is correct?
4. What is the significance of the discriminant in a quadratic equation?
5. How do the coefficients in a quadratic equation affect its graph?

Tip: Always check your solutions by substituting the $x$-values back into both original equations to ensure the points are valid intersections.