Question
Solve the equation log2(x2−12x+43)=3
log
2
(
𝑥
2
−
12
𝑥
+
43
)
=
3
.(1 point)
Responses
x= 6±i7–√
𝑥
=
6
±
𝑖
7
x equals 6 plus or minus i Start Root 7 End Root
x= 6±2–√
𝑥
=
6
±
2
x equals 6 plus or minus Start Root 2 End Root
x=5,7
𝑥
=
5
,
log
2
(
𝑥
2
−
12
𝑥
+
43
)
=
3
.(1 point)
Responses
x= 6±i7–√
𝑥
=
6
±
𝑖
7
x equals 6 plus or minus i Start Root 7 End Root
x= 6±2–√
𝑥
=
6
±
2
x equals 6 plus or minus Start Root 2 End Root
x=5,7
𝑥
=
5
,
Answers
GPT-4o mini
To solve the equation \( \log_2(x^2 - 12x + 43) = 3 \), we will first rewrite it in its exponential form:
\[
x^2 - 12x + 43 = 2^3
\]
Calculating the right side gives:
\[
x^2 - 12x + 43 = 8
\]
Next, we simplify the equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
This simplifies to:
\[
x^2 - 12x + 35 = 0
\]
Now, we will factor the quadratic:
\[
(x - 5)(x - 7) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
Thus, the solutions to the equation are:
\[
x = 5, \quad x = 7
\]
The appropriate response is:
\( x = 5, 7 \)
\[
x^2 - 12x + 43 = 2^3
\]
Calculating the right side gives:
\[
x^2 - 12x + 43 = 8
\]
Next, we simplify the equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
This simplifies to:
\[
x^2 - 12x + 35 = 0
\]
Now, we will factor the quadratic:
\[
(x - 5)(x - 7) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
Thus, the solutions to the equation are:
\[
x = 5, \quad x = 7
\]
The appropriate response is:
\( x = 5, 7 \)