Question

solve the equation \\(\log_{3}(x^{2} - 4x + 22) = 3\\). remember to check your solution(s). show all work to receive full credit. you may either type your work below, or you may click the picture icon to upload a picture of your work. (3 points)

Explanation:

Step1: Convert log to exponential form

Recall $\log_b(a)=c \implies b^c=a$. So:
$$3^3 = x^2 - 4x + 22$$

Step2: Simplify left-hand side

Calculate $3^3$:
$$27 = x^2 - 4x + 22$$

Step3: Rearrange to quadratic equation

Subtract 27 from both sides:
$$x^2 - 4x - 5 = 0$$

Step4: Factor the quadratic

Find two factors of -5 that sum to -4:
$$(x - 5)(x + 1) = 0$$

Step5: Solve for x

Set each factor equal to 0:
$x - 5 = 0 \implies x=5$; $x + 1 = 0 \implies x=-1$

Step6: Verify solutions

For $x=5$: $\log_3(5^2 - 4(5) + 22) = \log_3(25-20+22)=\log_3(27)=3$, which is valid.
For $x=-1$: $\log_3((-1)^2 - 4(-1) + 22) = \log_3(1+4+22)=\log_3(27)=3$, which is valid.

Answer:

$x=5$ and $x=-1$