The line y+x=9 intersects the curve y=x^2-2x+3 at the points P and Q .

(a) Find the coordinates of P and Q .
(b) Sketch the line and the curve on the same set of axes.
(c) Calculate the area enclosed by the line and the curve.

Set the two equations equal to each other and solve

Go look at your integration chapter - area bounded by two curve
Integration (line - parabola)dx
Then you are going to deal with adding the part of the parabola under the x-axis

To find the coordinates of the points of intersection between the line y + x = 9 and the curve y = x^2 - 2x + 3, we need to solve the system of equations formed by equating the two equations.

(a) Finding the coordinates of P and Q:

Step 1: Substitute y = 9 - x into the curve equation.
9 - x = x^2 - 2x + 3

Step 2: Rearrange the equation to get it in quadratic form.
x^2 - 2x + x - 6 = 0
x^2 - x - 6 = 0

Step 3: Factorize the quadratic equation or use the quadratic formula to find the values of x.
(x - 3)(x + 2) = 0
x - 3 = 0 or x + 2 = 0
x = 3 or x = -2

Step 4: Substitute the values of x back into either equation to find the corresponding y values.
For x = 3, y = 9 - 3 = 6.
For x = -2, y = 9 - (-2) = 11.

Therefore, the coordinates of P are (3, 6) and the coordinates of Q are (-2, 11).

(b) Sketching the line and the curve:

To sketch the line y + x = 9 and the curve y = x^2 - 2x + 3 on the same set of axes, follow these steps:

Step 1: Plot the points P and Q on the graph.

Step 2: Draw a straight line passing through the points P and Q.

Step 3: Plot additional points on the line, if needed, to create a clear representation.

Step 4: Sketch the curve y = x^2 - 2x + 3. To do this, you can plot additional points on the curve or use symmetry (since it is a quadratic function).

Step 5: Label the line as "y + x = 9" and the curve as "y = x^2 - 2x + 3" for clarity.

(c) Calculating the area enclosed by the line and the curve:

To calculate the area enclosed by the line and the curve, we need to find the points of intersection and calculate the definite integral of the difference between the curve and the line within that interval.

Step 1: Find the x-coordinates of the points of intersection by solving the system of equations as described in part (a).

Step 2: Determine the limits of integration. In this case, the limits of integration will be the x-coordinates of the points of intersection, which are x = 3 and x = -2.

Step 3: Set up the integral. The integral represents the area between the curve and the line. In this case, it would be ∫[(x^2 - 2x + 3) - (9 - x)] dx, with the limits of integration from x = -2 to x = 3.

Step 4: Evaluate the definite integral using integration techniques.