Question
Solve the equation log2(x2−12x+43)=3
.(1 point)
Responses
x=5,7
x equals 5 comma 7
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= 6±i7–√
.(1 point)
Responses
x=5,7
x equals 5 comma 7
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= 6±i7–√
Answers
GPT-4o mini
To solve the equation \( \log_2(x^2 - 12x + 43) = 3 \), we first convert from logarithmic form to exponential form.
The equation can be rewritten as:
\[
x^2 - 12x + 43 = 2^3
\]
Calculating \( 2^3 \):
\[
2^3 = 8
\]
Now we have the equation:
\[
x^2 - 12x + 43 = 8
\]
Next, we can rearrange this equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
This simplifies to:
\[
x^2 - 12x + 35 = 0
\]
Now we can factor this quadratic equation. We need two numbers that multiply to 35 and add up to -12. The numbers -5 and -7 fit these criteria. Thus, we can factor it as:
\[
(x - 5)(x - 7) = 0
\]
Setting each factor equal to zero gives the solutions:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
So the solutions are:
\[
x = 5, 7
\]
The correct response is:
**x equals 5 comma 7**
The equation can be rewritten as:
\[
x^2 - 12x + 43 = 2^3
\]
Calculating \( 2^3 \):
\[
2^3 = 8
\]
Now we have the equation:
\[
x^2 - 12x + 43 = 8
\]
Next, we can rearrange this equation:
\[
x^2 - 12x + 43 - 8 = 0
\]
This simplifies to:
\[
x^2 - 12x + 35 = 0
\]
Now we can factor this quadratic equation. We need two numbers that multiply to 35 and add up to -12. The numbers -5 and -7 fit these criteria. Thus, we can factor it as:
\[
(x - 5)(x - 7) = 0
\]
Setting each factor equal to zero gives the solutions:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
So the solutions are:
\[
x = 5, 7
\]
The correct response is:
**x equals 5 comma 7**