Question

Solve the equation log2(x2−12x+43)=3
.(1 point)
Responses

x=5,7
x equals 5 comma 7

x= 6±2–√
x equals 6 plus or minus Start Root 2 End Root

x= 6±i7–√
x equals 6 plus or minus i Start Root 7 End Root

x=−5,−7

Answer 1

To solve the equation \( \log_2(x^2 - 12x + 43) = 3 \), we first convert the logarithmic equation into its exponential form.

This gives us:

\[
x^2 - 12x + 43 = 2^3
\]

Calculating \( 2^3 \):

\[
x^2 - 12x + 43 = 8
\]

Next, we set the equation to zero by moving 8 to the left side:

\[
x^2 - 12x + 43 - 8 = 0
\]

This simplifies to:

\[
x^2 - 12x + 35 = 0
\]

Now we can factor the quadratic:

\[
(x - 5)(x - 7) = 0
\]

Setting each factor 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:

\[
x = 5, 7
\]

So the correct response is:

**x equals 5 comma 7**.