The problem provided asks you to solve the quadratic equation using the quadratic formula:

$x^2 - 6x + 9 = 0$

Step 1: Identify the coefficients

In the quadratic equation $ax^2 + bx + c = 0$, the coefficients are:

• $a = 1$
• $b = -6$
• $c = 9$

Step 2: Apply the quadratic formula

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values of $a$, $b$, and $c$:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(9)}}{2(1)}$

Step 3: Simplify the expression

First, calculate the discriminant:

$(-6)^2 - 4(1)(9) = 36 - 36 = 0$

Since the discriminant is 0, there is exactly one real solution. Now, calculate $x$:

$x = \frac{6 \pm \sqrt{0}}{2} = \frac{6 \pm 0}{2} = \frac{6}{2} = 3$

Final Answer:

$x = 3$

Let me know if you have any questions or need further details!

Follow-up Questions:

1. How does the discriminant determine the nature of the roots?
2. Can you solve a quadratic equation with a negative discriminant?
3. How would the quadratic formula change if the coefficients were different?
4. What is the geometric interpretation of the solution to this quadratic equation?
5. How can you verify the solution by factoring the quadratic expression?

Tip: Always check the discriminant first to anticipate the number and type of roots (real or complex) before solving a quadratic equation.