Steps
So firstly, we have to make all the denominators the same. To do this, we need to find their LCD, or lowest common denominator. In this case, the LCD is gonna be 6(x + 3). With each fraction, multiply the numerator and denominator by the term that will get the denominator to the LCD:
[tex]\frac{-3}{x+3}*\frac{6}{6}=\frac{-18}{6(x+3)}\\\\\frac{1}{2}*\frac{3(x+3)}{3(x+3)}=\frac{3(x+3)}{6(x+3)}\\\\\frac{x}{6}*\frac{x+3}{x+3}=\frac{x(x+3)}{6(x+3)}\\\\\frac{-1}{2}*\frac{3(x+3)}{3(x+3)}=\frac{-3(x+3)}{6(x+3)}\\\\\\\\\frac{-18}{6(x+3)}+\frac{3(x+3)}{6(x+3)}=\frac{x(x+3)}{6(x+3)}+\frac{-3(x+3)}{6(x+3)}[/tex]
Now with all the denominators the same, we can cancel them out by multiplying both sides by 6(x + 3):
[tex]-18+3(x+3)=x(x+3)+-3(x+3)[/tex]
Next, solve the multiplications:
[tex]-18+3x+9=x^2+3x-3x-9[/tex]
Next, combine like terms:
[tex]3x-9=x^2-9[/tex]
Next, subtract both sides by 3x and add both sides by 9:
[tex]0=x^2-3x[/tex]
Next, we want to make the right side of the equation a perfect square. To find the constant of the soon-to-be perfect square, divide the x coefficient by 2 and square the quotient. Add the result onto both sides of the equation:
[tex]-3\div 2=-1.5\\(-1.5)^2=2.25\\\\2.25=x^2-3x+2.25[/tex]
Now, factor the right side:
[tex]2.25=(x-1.5)^2[/tex]
Next, square root both sides of the equation:
[tex]\pm\ 1.5=x-1.5[/tex]
Next, add both sides by 1.5:
[tex]1.5\pm 1.5=x[/tex]
Lastly, solve the left side twice - once with the plus sign and once with the minus sign:
[tex]3,0=x[/tex]
Answer
In short, x = 0 and 3.
