Solve each rational equation. State the non-permissive values.
12/3๐ฅ+6โ10/5๐ฅ+10=3๐ฅ+1/2+3^2
To solve the rational equation, we first need to find the LCD (Least Common Denominator) of the fractions on both sides of the equation.
The denominators on the left side are 3x+6 and 5x+10. The LCD of these two terms is 30(x+2).
The denominator on the right side is 2, so the LCD for the entire equation is 30(x+2).
Next, we multiply every term in the equation by the LCD to eliminate the fractions:
(30(x+2))(12/3x + 6) - (30(x+2))(10/5x + 10) = (30(x+2))(3x+1/2) + (30(x+2))(3^2)
Simplifying this equation gives us:
40(x+2) - 60(x+2) = 15x^2 + 10(x+2) + 90(x+2)
Now, we distribute and combine like terms:
40x + 80 - 60x - 120 = 15x^2 + 10x + 20 + 90x + 180
Combining like terms again:
-20x - 40 = 15x^2 + 100x + 200
Rearranging the equation to set it equal to zero:
15x^2 + 120x + 240 = 0
Now, we can solve this quadratic equation. However, before doing that, we need to find the non-permissible values. These are the values of x that would make any of the denominators zero. In this case, we have 3x+6 and 5x+10.
Setting each denominator equal to zero and solving for x:
3x+6 = 0
3x = -6
x = -2
and
5x+10 = 0
5x = -10
x = -2
So, the non-permissible value for this equation is x = -2.
Now, we can solve the quadratic equation:
15x^2 + 120x + 240 = 0
Dividing each term by 15 to simplify:
x^2 + 8x + 16 = 0
Factoring the quadratic equation:
(x + 4)(x + 4) = 0
Setting each factor equal to zero and solving for x:
x + 4 = 0
x = -4
Therefore, the solution to the rational equation is x = -4.