The given equation is x2−6x+9=0. We can rewrite this equation in vertex form by completing the square.
x2−6x+9=(x−3)2=0
The vertex form of a parabola is y=a(x−h)2+k, where (h,k) is the vertex. In our case, a=1, h=3, and k=0. Thus, the vertex is (3,0).
We can choose x values on either side of the vertex (x=3) and calculate the corresponding y values.
- x=1: y=(1−3)2=(−2)2=4 So the point is (1,4).
- x=2: y=(2−3)2=(−1)2=1 So the point is (2,1).
- x=4: y=(4−3)2=(1)2=1 So the point is (4,1).
- x=5: y=(5−3)2=(2)2=4 So the point is (5,4).
Plot the vertex (3,0) and the four additional points: (1,4), (2,1), (4,1), and (5,4). Draw a smooth curve through these points to form the parabola.
The solutions to the equation x2−6x+9=0 are the x-values where the parabola intersects the x-axis (y=0). In this case, the parabola touches the x-axis at only one point: (3,0).