Solve the logarithmic quadratic: If log10(x^2 − 6x + 45) = 2, find all real values of x.

Introduction / Context:
We are given a base-10 logarithm of a quadratic expression equal to 2. Use the definition of logarithms to convert to an exponential equation and solve the resulting quadratic. Finally, confirm that the argument of the logarithm is positive for each solution (domain check).

Given Data / Assumptions:

• log10(x^2 − 6x + 45) = 2
• Domain requirement: x^2 − 6x + 45 > 0

Concept / Approach:
Use a^b = 10^2 when log10(a) = 2. So set the quadratic equal to 100 and solve. Because the resulting quadratic will have real roots, check that each keeps the log argument positive (which it does here since the expression equals 100 at the roots and is a parabola opening upward).

Step-by-Step Solution:
x^2 − 6x + 45 = 10^2 = 100x^2 − 6x − 55 = 0Discriminant: 36 + 220 = 256 ⇒ √256 = 16x = (6 ± 16)/2 ⇒ x = 11 or x = −5

Verification / Alternative check:
At x = 11: x^2 − 6x + 45 = 121 − 66 + 45 = 100 ⇒ log10(100) = 2. At x = −5: 25 + 30 + 45 = 100 ⇒ log10(100) = 2. Domain satisfied (argument = 100 > 0).

Why Other Options Are Wrong:
6,9 and 10,5 do not satisfy the transformed quadratic; “Only 11” ignores the second valid root; 9, −5 has 9 incorrect.

Common Pitfalls:
Failing to enforce the log domain or miscomputing the discriminant. Always convert log equations carefully and check positivity.

Final Answer:
11, −5