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:

$$x-5=0\Rightarrow x=5$$ $$x-7=0\Rightarrow x=7$$

Thus, the solutions to the equation are:

$$x=5,7$$

So the correct response is:

x equals 5 comma 7.