Solve the equation log2(x2−12x+43)=3

Question

.(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.