Solve each rational equation. State the non-permissive values.
12/3๐ฅ+6โ10/5๐ฅ+10=3๐ฅ+12+3๐ฅ^2
To solve the rational equation 12/(3๐ฅ+6)โ10/(5๐ฅ+10)=3๐ฅ+12+(3๐ฅ^2), we start by simplifying the equation.
Step 1: Simplify the left side of the equation by finding a common denominator for 12/(3๐ฅ+6) and -10/(5๐ฅ+10).
The common denominator for 12/(3๐ฅ+6) and -10/(5๐ฅ+10) is (3๐ฅ+6)(5๐ฅ+10).
12/(3๐ฅ+6) can be written as (12(5๐ฅ+10))/(3๐ฅ+6)(5๐ฅ+10).
-10/(5๐ฅ+10) can be written as (-10(3๐ฅ+6))/(3๐ฅ+6)(5๐ฅ+10).
So the left side of the equation becomes (12(5๐ฅ+10))/(3๐ฅ+6)(5๐ฅ+10) - (10(3๐ฅ+6))/(3๐ฅ+6)(5๐ฅ+10).
Simplifying this further, we get (60๐ฅ+120-30๐ฅ-60)/(3๐ฅ+6)(5๐ฅ+10), which becomes (30๐ฅ+60)/(3๐ฅ+6)(5๐ฅ+10).
Step 2: Simplify the right side of the equation by combining like terms.
The right side of the equation is 3๐ฅ+12+3๐ฅ^2, which cannot be further simplified.
Therefore, the equation becomes (30๐ฅ+60)/(3๐ฅ+6)(5๐ฅ+10) = 3๐ฅ+12+3๐ฅ^2.
Step 3: Find the non-permissible values by setting the denominators equal to zero and solving for ๐ฅ.
(3๐ฅ+6) = 0 or (5๐ฅ+10) = 0
Solving these equations, we get ๐ฅ = -2 and ๐ฅ = -2.
Therefore, the non-permissible values for this rational equation are ๐ฅ = -2.
Step 4: Solve the equation by cross-multiplying and simplifying.
Multiply both sides of the equation by (3๐ฅ+6)(5๐ฅ+10) to eliminate the denominators.
(30๐ฅ+60)(3๐ฅ+6) = (3๐ฅ+12+3๐ฅ^2)(3๐ฅ+6)(5๐ฅ+10).
Expanding both sides of the equation, we get:
90๐ฅ^2 + 180๐ฅ + 180๐ฅ + 360 = (3๐ฅ+12+3๐ฅ^2)(15๐ฅ^2+90๐ฅ +50๐ฅ+300).
Simplifying further, we get:
90๐ฅ^2 + 360๐ฅ + 360 = 45๐ฅ^3 + 450๐ฅ^2 + 810๐ฅ + 3600.
Rearranging the terms, we get:
45๐ฅ^3 + 450๐ฅ^2 + 720๐ฅ + 3240 = 90๐ฅ^2 + 360๐ฅ.
Combining like terms, we get:
45๐ฅ^3 + 360๐ฅ^2 + 720๐ฅ + 3240 = 0.
Step 5: Solve the equation 45๐ฅ^3 + 360๐ฅ^2 + 720๐ฅ + 3240 = 0.
Since this equation is a cubic equation, it cannot be solved algebraically. It must be solved using numerical methods such as factoring, the Rational Root Theorem, or approximate methods like Newton-Raphson.
Without further information or specific instructions, we cannot provide the exact solution for this cubic equation.