Problem Analysis

Let's break down the tasks step-by-step:

a) Solve the equation:

$-x^2 + 4x - 7 = -4x + 3$

b) Plot the parabola and the line:

The parabola is given by: $y = -x^2 + 4x - 7$ The line is given by: $y = -4x + 3$

c) Provide a graphical interpretation of the solutions from part (a).

Solution for Part (a)

To solve the equation: $-x^2 + 4x - 7 = -4x + 3$

1. Rearrange all terms to one side of the equation: $-x^2 + 4x - 7 + 4x - 3 = 0$

2. Simplify the equation: $-x^2 + 8x - 10 = 0$

3. Multiply the entire equation by -1 for simplicity: $x^2 - 8x + 10 = 0$

4. Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -8$, and $c = 10$.

5. Plug the values into the formula: $x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$ $x = \frac{8 \pm \sqrt{64 - 40}}{2}$ $x = \frac{8 \pm \sqrt{24}}{2}$ $x = \frac{8 \pm 2\sqrt{6}}{2}$ $x = 4 \pm \sqrt{6}$

Thus, the solutions are: $x_1 = 4 + \sqrt{6}, \quad x_2 = 4 - \sqrt{6}$

Solution for Part (b)

I will describe how to plot the graph:

1. Plot the parabola $y = -x^2 + 4x - 7$.
2. Plot the line $y = -4x + 3$ on the same coordinate system.

Solution for Part (c)

The solutions $x_1 = 4 + \sqrt{6}$ and $x_2 = 4 - \sqrt{6}$ represent the x-coordinates where the parabola and the line intersect. Graphically, these points of intersection are where the solutions to the equation $-x^2 + 4x - 7 = -4x + 3$ occur.

Would you like me to provide a detailed graph for visualization? Or is there another part you'd like more information about?

Related Questions:

1. What is the significance of the solutions $x_1$ and $x_2$ in terms of the graph?
2. How would the solutions change if we altered the coefficients in the parabola's equation?
3. What happens to the intersection points if the line equation changes?
4. Can you find the vertex of the parabola $y = -x^2 + 4x - 7$?
5. How do the discriminant values of a quadratic equation influence the type of solutions?

Tip:

When solving a quadratic equation graphically, the solutions correspond to the x-coordinates where the two graphs intersect. This can help in understanding the nature of the solutions (real or complex).