In the xy-plane, the graphs of the given equations intersect at two points. What is the sum of the x-coordinates of the…

To find the points of intersection, we can set the expressions for y equal to each other:

x² - 4x + 5 = x + 1

Next, we can rewrite this equation in the form ax² + bx + c = 0 by moving all terms to one side:

x² - 4x - x + 5 - 1 = 0

x² - 5x + 4 = 0

The x-coordinates of the intersection points are the solutions to this quadratic equation. We can find the sum of the solutions in two ways.

Method 1: Factor the quadratic equation.

(x - 1)(x - 4) = 0

The solutions are x = 1 and x = 4. The sum of the x-coordinates is 1 + 4 = 5.

Method 2: Use the formula for the sum of the roots.

For a quadratic equation in the form ax² + bx + c = 0, the sum of the roots is -b/a. In the equation x² - 5x + 4 = 0, we have a = 1, b = -5, and c = 4.

Therefore, the sum of the roots is -(-5)/1 = 5.

The correct answer is 5.

A is incorrect. This is the result of a sign error when using the formula for the sum of roots of a quadratic equation (calculating b/a instead of -b/a).

C is incorrect. This is the sum of the y-coordinates of the intersection points. When x=1, y=1+1=2. When x=4, y=4+1=5. The sum of the y-coordinates is 2 + 5 = 7.

D is incorrect. This is one of the two x-coordinates of the intersection points, but the question asks for the sum of both x-coordinates.