Question

Does the equation x2 - 4x + y2 = -3 intersect the x-axis?

Answer 1

Yes, because the center is on the x-axis.

Explanation:

First, write the equation in standard form by completing the square.

x2 - 4x + y2 = -3

x2 - 4x + 4 + y2 = -3 + 4

(x - 2)2 + y2 = 1

The circle is centered at (2, 0) with a radius of 1. Since the circle is centered on the x-axis, it intersects the x-axis two times, at (3, 0) and (1, 0).

Answer 2

To find out whether or not the equation x^2 - 4x + y^2 = -3 intersects the x-axis, we must set y = 0 in the equation (because at every point on the x-axis, y = 0).

x^2 - 4x + 0 = -3

We then want to solve for x. We can do this by factoring.

x^2 - 4x + 3 = 0

By factoring...

(x - 3)(x - 1)

We can set each of these equations = 0 to solve where the function crosses the x-axis.

x - 3 = 0
x = 3

x - 1 = 0
x = 1

So we know at x = 1 and x = 3, the function x^2 - 4x + y^2 = -3 intersects the x-axis.