Solving A Tricky Rational Equation: A Step-by-Step Guide
Alright guys, let's dive into solving this equation: (2x)/(4x^2+3x+8) + (3x)/(4x^2-6x+8) = 1/6. This looks like a beast, but don't worry, we'll break it down piece by piece. We're dealing with a rational equation here, which means we've got fractions with polynomials. The trick is to manipulate it strategically to get rid of those fractions and end up with something we can actually solve. Buckle up; it's going to be a fun ride!
Initial Assessment and Simplification
Before we jump into any heavy lifting, let's take a good look at the equation. We have (2x)/(4x^2+3x+8) + (3x)/(4x^2-6x+8) = 1/6. Notice that 'x' appears in both numerators. This is a good sign because it might allow us to simplify things early on. Specifically, let's see if we can factor out 'x' from anywhere to make the denominators easier to handle. Factoring out 'x' from the left side numerators can make this problem more approachable. We have:
2x / (4x^2 + 3x + 8) + 3x / (4x^2 - 6x + 8) = 1/6
Now, let's consider a substitution to make the equation look cleaner. Divide both numerators and denominators by 'x' (assuming x ≠ 0). This gives us:
2 / (4x + 3 + 8/x) + 3 / (4x - 6 + 8/x) = 1/6
This might not seem like much, but it sets us up for a clever substitution that will simplify the equation dramatically. Keep your eyes peeled; the magic is coming!
Making a Smart Substitution
Okay, now comes the clever part. Look at the denominators: 4x + 3 + 8/x and 4x - 6 + 8/x. They both have 4x + 8/x in them. Let's make a substitution: Let y = 4x + 8/x. This will transform our equation into something much more manageable.
Substituting y = 4x + 8/x, our equation becomes:
2 / (y + 3) + 3 / (y - 6) = 1/6
See how much simpler that looks? Now we have an equation with just one variable, 'y'. This is a huge step forward. We've gone from a complicated rational equation to a simpler one that we can solve using standard algebraic techniques. Now, let's get rid of those fractions!
Clearing the Fractions and Solving for 'y'
To get rid of the fractions, we need to multiply both sides of the equation by the least common denominator (LCD). In this case, the LCD is 6 * (y + 3) * (y - 6). Multiplying both sides by the LCD gives us:
6 * (y + 3) * (y - 6) * [2 / (y + 3) + 3 / (y - 6)] = 6 * (y + 3) * (y - 6) * (1/6)
Simplifying, we get:
12 * (y - 6) + 18 * (y + 3) = (y + 3) * (y - 6)
Expanding and simplifying further:
12y - 72 + 18y + 54 = y^2 - 3y - 18
Combining like terms:
30y - 18 = y^2 - 3y - 18
Now, let's move everything to one side to get a quadratic equation:
y^2 - 33y = 0
Factoring out 'y':
y * (y - 33) = 0
This gives us two possible solutions for 'y':
y = 0 or y = 33
Great! We've found the values of 'y'. But remember, we're trying to solve for 'x'. So, we need to substitute back and solve for 'x' in each case.
Substituting Back and Solving for 'x'
We have two cases to consider:
Case 1: y = 0
Recall that y = 4x + 8/x. Substituting y = 0, we get:
4x + 8/x = 0
Multiplying by 'x' to clear the fraction:
4x^2 + 8 = 0
4x^2 = -8
x^2 = -2
Since we're looking for real solutions, and we can't have a negative number under a square root, there are no real solutions in this case. These would be imaginary solutions.
Case 2: y = 33
Substituting y = 33 into y = 4x + 8/x, we get:
4x + 8/x = 33
Multiplying by 'x' to clear the fraction:
4x^2 + 8 = 33x
Rearranging into a quadratic equation:
4x^2 - 33x + 8 = 0
Now we need to solve this quadratic equation. We can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 4, b = -33, and c = 8.
Plugging in the values:
x = (33 ± √((-33)^2 - 4 * 4 * 8)) / (2 * 4)
x = (33 ± √(1089 - 128)) / 8
x = (33 ± √961) / 8
x = (33 ± 31) / 8
This gives us two possible solutions for 'x':
x = (33 + 31) / 8 = 64 / 8 = 8
x = (33 - 31) / 8 = 2 / 8 = 1/4
So, we have two real solutions for 'x': x = 8 and x = 1/4.
Verification and Conclusion
It's always a good idea to check our solutions by plugging them back into the original equation to make sure they work. Let's do that:
For x = 8:
(2*8) / (4*(8^2) + 3*8 + 8) + (3*8) / (4*(8^2) - 6*8 + 8) = 1/6
16 / (256 + 24 + 8) + 24 / (256 - 48 + 8) = 1/6
16 / 288 + 24 / 216 = 1/6
1 / 18 + 1 / 9 = 1/6
1 / 18 + 2 / 18 = 1/6
3 / 18 = 1/6
1/6 = 1/6 (This solution checks out!)
For x = 1/4:
(2*(1/4)) / (4*(1/4)^2 + 3*(1/4) + 8) + (3*(1/4)) / (4*(1/4)^2 - 6*(1/4) + 8) = 1/6
(1/2) / (1/4 + 3/4 + 8) + (3/4) / (1/4 - 3/2 + 8) = 1/6
(1/2) / (1 + 8) + (3/4) / (1/4 - 6/4 + 32/4) = 1/6
(1/2) / 9 + (3/4) / (27/4) = 1/6
1 / 18 + 3 / 27 = 1/6
1 / 18 + 1 / 9 = 1/6
1 / 18 + 2 / 18 = 1/6
3 / 18 = 1/6
1/6 = 1/6 (This solution also checks out!)
Therefore, the solutions to the equation (2x)/(4x^2+3x+8) + (3x)/(4x^2-6x+8) = 1/6 are x = 8 and x = 1/4. We started with a complex rational equation, used a clever substitution to simplify it, solved for the new variable, substituted back to find 'x', and verified our solutions. Great job, guys! We conquered that beast of an equation!
