Solving The Equation (8r + 2)/5 = 2/r A Step-by-Step Guide
In this article, we will delve into the process of solving the equation (8r + 2)/5 = 2/r. This equation falls under the category of rational equations, which involve fractions with variables in the denominator. Solving such equations requires a systematic approach to eliminate the fractions and arrive at a solution for the variable 'r'. We will explore the steps involved, the underlying mathematical principles, and potential pitfalls to avoid.
Understanding the Equation: (8r + 2)/5 = 2/r
Before we dive into the solution, let's break down the equation (8r + 2)/5 = 2/r. We have a rational expression on both sides of the equation. The left side consists of the fraction (8r + 2) divided by 5, while the right side is simply 2 divided by 'r'. Our goal is to find the value(s) of 'r' that make this equation true. Importantly, we need to consider that 'r' cannot be equal to zero, as this would lead to division by zero on the right side, which is undefined in mathematics. This constraint is crucial and will influence our solution process.
The presence of fractions in an equation can often complicate the solving process. Therefore, our initial strategy will be to eliminate these fractions. This is typically achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 5 and 'r'. The LCM of 5 and 'r' is simply 5r, as they share no common factors other than 1. Multiplying both sides by 5r will clear the fractions, transforming the equation into a more manageable form.
Step-by-Step Solution
1. Eliminating Fractions
To begin, we multiply both sides of the equation (8r + 2)/5 = 2/r by the least common multiple (LCM) of the denominators, which is 5r:
5r * [(8r + 2)/5] = 5r * (2/r)
On the left side, the 5 in the denominator cancels with the 5 we multiplied by, leaving us with:
r * (8r + 2)
On the right side, the 'r' in the denominator cancels with the 'r' we multiplied by, leaving us with:
5 * 2
So, after eliminating the fractions, our equation becomes:
r(8r + 2) = 10
2. Expanding and Rearranging
Next, we expand the left side of the equation by distributing the 'r':
8r^2 + 2r = 10
Now, to solve for 'r', we need to rearrange the equation into a standard quadratic form, which is ax^2 + bx + c = 0. To do this, we subtract 10 from both sides:
8r^2 + 2r - 10 = 0
3. Simplifying the Quadratic Equation
We can simplify the quadratic equation by dividing all terms by their greatest common divisor, which is 2:
(8r^2 + 2r - 10) / 2 = 0 / 2
This simplifies to:
4r^2 + r - 5 = 0
Now we have a simplified quadratic equation in the standard form.
4. Solving the Quadratic Equation
To solve the quadratic equation 4r^2 + r - 5 = 0, we can use several methods: factoring, completing the square, or the quadratic formula. In this case, factoring is a viable option.
We are looking for two numbers that multiply to (4 * -5) = -20 and add up to 1 (the coefficient of the 'r' term). These numbers are 5 and -4.
We can rewrite the middle term (r) as 5r - 4r:
4r^2 + 5r - 4r - 5 = 0
Now, we can factor by grouping:
r(4r + 5) - 1(4r + 5) = 0
This gives us:
(r - 1)(4r + 5) = 0
Setting each factor equal to zero gives us the possible solutions for 'r':
r - 1 = 0 or 4r + 5 = 0
Solving for 'r' in each case:
r = 1 or r = -5/4
5. Verifying the Solutions
It's crucial to verify our solutions by plugging them back into the original equation to ensure they are valid and do not lead to any undefined operations (like division by zero). Our original equation was (8r + 2)/5 = 2/r.
Case 1: r = 1
Substituting r = 1 into the equation:
(8(1) + 2)/5 = 2/1
(8 + 2)/5 = 2
10/5 = 2
2 = 2
This solution is valid.
Case 2: r = -5/4
Substituting r = -5/4 into the equation:
(8(-5/4) + 2)/5 = 2/(-5/4)
(-10 + 2)/5 = 2 * (-4/5)
-8/5 = -8/5
This solution is also valid.
6. Final Solutions
Therefore, the solutions to the equation (8r + 2)/5 = 2/r are r = 1 and r = -5/4.
Common Mistakes and Pitfalls
When solving rational equations, there are a few common mistakes to watch out for:
- Forgetting to Check for Extraneous Solutions: As demonstrated in our solution process, it's essential to substitute the obtained solutions back into the original equation. This step helps identify extraneous solutions, which are values that satisfy the transformed equation but not the original one. Extraneous solutions often arise when we square both sides of an equation or multiply by an expression containing the variable. In our example, we were fortunate that both solutions were valid, but this might not always be the case.
- Dividing by Zero: One of the most critical considerations when working with rational expressions is to avoid division by zero. Before solving, identify any values of the variable that would make the denominator zero. These values must be excluded from the solution set. In our equation, (8r + 2)/5 = 2/r, we noted that r cannot be zero. This is a crucial restriction to keep in mind throughout the solving process.
- Incorrectly Applying the Distributive Property: When eliminating fractions, we often need to multiply an expression by a term containing the variable. It's vital to apply the distributive property correctly, ensuring that each term within the parentheses is multiplied by the outside term. For example, in our equation, we multiplied 5r by (8r + 2)/5. We had to make sure that 5r was multiplied by both 8r and 2 after the denominator was cancelled out.
- Errors in Factoring or Applying the Quadratic Formula: If the equation leads to a quadratic form, accurately factoring the quadratic expression or correctly using the quadratic formula is essential. Mistakes in these steps can lead to incorrect solutions. Always double-check your factoring and ensure you've plugged the values into the quadratic formula correctly.
- Not Multiplying All Terms by the LCM: When clearing fractions, it is imperative to multiply every term in the equation by the least common multiple (LCM). For instance, in our example, we multiplied both sides of the equation (8r + 2)/5 = 2/r by 5r. This means we multiplied both the term on the left side, (8r + 2)/5, and the term on the right side, 2/r, by 5r. Forgetting to multiply any term by the LCM will result in an unbalanced equation and incorrect solutions.
By being mindful of these potential pitfalls and diligently following the steps outlined in our comprehensive solution, you can confidently tackle rational equations and find accurate solutions.
Alternative Methods for Solving Quadratic Equations
While we solved the quadratic equation 4r^2 + r - 5 = 0 by factoring, it's important to be aware of alternative methods that can be used, especially when factoring is not straightforward.
1. The Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is:
r = [-b ± √(b^2 - 4ac)] / (2a)
In our case, a = 4, b = 1, and c = -5. Substituting these values into the quadratic formula gives us:
r = [-1 ± √(1^2 - 4 * 4 * -5)] / (2 * 4)
r = [-1 ± √(1 + 80)] / 8
r = [-1 ± √81] / 8
r = [-1 ± 9] / 8
This gives us two solutions:
r = (-1 + 9) / 8 = 8 / 8 = 1
r = (-1 - 9) / 8 = -10 / 8 = -5/4
As we can see, the quadratic formula yields the same solutions we obtained through factoring.
2. Completing the Square
Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side.
Starting with our equation 4r^2 + r - 5 = 0, we first divide by 4 to make the coefficient of r^2 equal to 1:
r^2 + (1/4)r - 5/4 = 0
Next, we move the constant term to the right side:
r^2 + (1/4)r = 5/4
To complete the square, we take half of the coefficient of the 'r' term (which is 1/4), square it ((1/4) / 2 = 1/8, (1/8)^2 = 1/64), and add it to both sides:
r^2 + (1/4)r + 1/64 = 5/4 + 1/64
Now, the left side is a perfect square trinomial:
(r + 1/8)^2 = 81/64
Taking the square root of both sides:
r + 1/8 = ±√(81/64)
r + 1/8 = ±9/8
Solving for 'r':
r = -1/8 ± 9/8
This gives us two solutions:
r = (-1/8 + 9/8) = 8/8 = 1
r = (-1/8 - 9/8) = -10/8 = -5/4
Again, we arrive at the same solutions as with factoring and the quadratic formula. Understanding these alternative methods provides flexibility in solving quadratic equations, allowing you to choose the approach that best suits the specific equation you are facing.
Conclusion
In this comprehensive guide, we have demonstrated how to solve the equation (8r + 2)/5 = 2/r step-by-step. We began by eliminating fractions through multiplication by the least common multiple, then simplified the resulting quadratic equation. We successfully solved for 'r' using factoring and verified our solutions. Additionally, we highlighted common mistakes to avoid and explored alternative methods for solving quadratic equations, such as the quadratic formula and completing the square. By mastering these techniques, you can confidently approach and solve a wide range of rational equations.
