Question
Solve the equation log2(x2−12x+43)=3
.(1 point)
Responses
x=−5,−7
x equals negative 5 comma negative 7
x= 6±2–√
x equals 6 plus or minus Start Root 2 End Root
x=5,7
x equals 5 comma 7
x= 6±i7–√
.(1 point)
Responses
x=−5,−7
x equals negative 5 comma negative 7
x= 6±2–√
x equals 6 plus or minus Start Root 2 End Root
x=5,7
x equals 5 comma 7
x= 6±i7–√
Answers
GPT-4o mini
To solve the equation \( \log_2(x^2 - 12x + 43) = 3 \), we first rewrite the logarithmic equation in its exponential form:
\[
x^2 - 12x + 43 = 2^3
\]
Calculating \( 2^3 \) gives us:
\[
x^2 - 12x + 43 = 8
\]
Now, we can rearrange the equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
Simplifying further:
\[
x^2 - 12x + 35 = 0
\]
Next, we can factor the quadratic equation:
\[
(x - 5)(x - 7) = 0
\]
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:
\[
x = 5, 7.
\]
Therefore, the correct response is:
**x=5,7**.
\[
x^2 - 12x + 43 = 2^3
\]
Calculating \( 2^3 \) gives us:
\[
x^2 - 12x + 43 = 8
\]
Now, we can rearrange the equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
Simplifying further:
\[
x^2 - 12x + 35 = 0
\]
Next, we can factor the quadratic equation:
\[
(x - 5)(x - 7) = 0
\]
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:
\[
x = 5, 7.
\]
Therefore, the correct response is:
**x=5,7**.