Questions: Solve the equation by graphing. x^2-6x+9=0 First, graph the associated parabola by plotting the vertex and four additional points, two on each side of the vertex. Then, use the graph to give the solution(s) to the equation. If there is more than one solution, separate them with commas. Solution(s): x=

Solution Steps

The given equation is \(x^2 - 6x + 9 = 0\). We can rewrite this equation in vertex form by completing the square.

\(x^2 - 6x + 9 = (x-3)^2 = 0\)

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 \(x^2 - 6x + 9 = 0\) 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: \((3,0)\).

Final Answer