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