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