Answer
$9$ miles per hour.
Work Step by Step
Let the speed of the boat in still water be $=x$ miles per hour.
The speed of the current $=3$ miles per hour.
Relative speed of the boat against the current $=x-3$.
Relative speed of the boat with the current $=x+3$.
Distance travelled both sides $12$ miles.
Formula the time is $Time =\frac{Distance}{Speed}$.
Time taken against the current $=\frac{12}{x-3}$
Time taken with the current $=\frac{12}{x+3}$
Total time $=3$ hours.
$\Rightarrow 3=\frac{12}{x-3}+\frac{12}{x+3}$
Multiply both sides by the Least Common Denominator $(x-3)(x+3)$ to clear fractions.
$\Rightarrow 3(x-3)(x+3)=(x-3)(x+3)\left (\frac{12}{x-3}+\frac{12}{x+3}\right)$
Use the distributive property.
$\Rightarrow 3(x-3)(x+3)=(x-3)(x+3)\left (\frac{12}{x-3}\right)+(x-3)(x+3)\left (\frac{12}{x+3}\right)$
Cancel common factors.
$\Rightarrow 3(x-3)(x+3)=12(x+3)+12(x-3)$
Use the special formula $A^2-B^2=(A+B)(A-B)$ and the distributive property.
$\Rightarrow 3(x^2-9)=12x+36+12x-36$
Simplify.
$\Rightarrow 3(x^2-9)=24x$
Divide both sides by $3$.
$\Rightarrow \frac{3(x^2-9)}{3}=\frac{24x}{3}$
Simplify.
$\Rightarrow x^2-9=8x$
Add $-8x$ to both sides.\
$\Rightarrow x^2-9-8x=8x-8x$
Simplify.
$\Rightarrow x^2-8x-9=0$
Rewrite the middle term $-8x$ as $-9x+1x$.
$\Rightarrow x^2-9x+1x-9=0$
Group the terms.
$\Rightarrow (x^2-9x)+(1x-9)=0$
Factor each group.
$\Rightarrow x(x-9)+1(x-9)=0$
Factor out $(x-9)$.
$\Rightarrow (x-9)(x+1)=0$
By using zero product rule set each factor equal to zero.
$\Rightarrow x-9=0$ or $x+1=0$
Isolate $x$.
$\Rightarrow x=9$ or $x=-1$
Take positive value.
$x=9$
Hence, the boat's rate in still water $9$ miles per hour.
Note: The equation is defined for all real values of $x$ except the zeros of the denominators which are $-3$ and $3$. Since the solution we found does not represent a zero of a fraction, it means it is correct.
