10) \( x^{2}-6 x+9>0 \)

Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x^{2}-6x+9=0\) - step1: Factor the expression: \(\left(x-3\right)^{2}=0\) - step2: Simplify the expression: \(x-3=0\) - step3: Move the constant to the right side: \(x=0+3\) - step4: Remove 0: \(x=3\) To solve the inequality \( x^{2}-6x+9>0 \), we first need to find the roots of the corresponding equation \( x^{2}-6x+9=0 \). 1. **Finding the roots:** The equation can be factored as: \[ (x-3)^2 = 0 \] This gives us a double root at \( x = 3 \). 2. **Analyzing the inequality:** The expression \( x^{2}-6x+9 \) can be rewritten as \( (x-3)^2 \). Now, we need to determine where \( (x-3)^2 > 0 \): - The expression \( (x-3)^2 \) is equal to 0 at \( x = 3 \). - For all other values of \( x \) (i.e., \( x < 3 \) or \( x > 3 \)), \( (x-3)^2 \) is positive. 3. **Conclusion:** Therefore, the solution to the inequality \( x^{2}-6x+9>0 \) is: \[ x < 3 \quad \text{or} \quad x > 3 \] In interval notation, this can be expressed as: \[ (-\infty, 3) \cup (3, \infty) \]