Let's solve the equation $x^2 - 6x + 9 = 25$.

Step 1: Bring the equation to standard form

Subtract 25 from both sides: $x^2 - 6x + 9 - 25 = 0$ $x^2 - 6x - 16 = 0$

Step 2: Apply the quadratic formula

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -6$, and $c = -16$.

Step 3: Calculate the discriminant

$b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100$

Step 4: Substitute into the formula

$x = \frac{-(-6) \pm \sqrt{100}}{2(1)}$ $x = \frac{6 \pm 10}{2}$

Step 5: Simplify

$x = \frac{6 + 10}{2} = \frac{16}{2} = 8$ $x = \frac{6 - 10}{2} = \frac{-4}{2} = -2$

Final Answer:

$x = 8 \quad \text{or} \quad x = -2$

Would you like a detailed verification or have any questions?

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Here are five related questions to explore:

1. How do you derive the quadratic formula from completing the square?
2. What happens if the discriminant is negative?
3. How does completing the square solve this problem differently?
4. What are real-world applications of quadratic equations?
5. Can this problem be factored instead of using the formula?

Tip: Always check your solutions by substituting them back into the original equation.