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How to Solve the Equation x/(x-1) - 1/(x-2) = 0: A Step-by-Step Guide
Solving rational equations, which are equations containing fractions with polynomials in the denominator, is a fundamental skill in algebra. Consider this: one common example is the equation x/(x-1) - 1/(x-2) = 0. This article provides a detailed, step-by-step guide on how to solve this specific equation, explaining the underlying principles and highlighting common pitfalls to avoid. By the end, you will have a clear methodology to tackle similar problems confidently Turns out it matters..
Introduction: Understanding the Problem
The equation we are tasked with solving is:
x/(x-1) - 1/(x-2) = 0
At first glance, this looks like a simple algebraic problem. Still, the presence of variables in the denominators introduces an important constraint: the denominators cannot be equal to zero, as division by zero is undefined. Put another way, for our equation:
- x - 1 ≠ 0, which implies x ≠ 1
- x - 2 ≠ 0, which implies x ≠ 2
These values, x = 1 and x = 2, are excluded from the possible solutions before we even begin. We will need to keep this in mind and check our final answer against these restrictions Worth knowing..
The goal is to find the value(s) of x that make the equation true. Our strategy will be to eliminate the fractions by finding a common denominator, simplifying the equation into a more familiar form (a linear equation), solving for x, and finally verifying the solution.
Step 1: Find the Least Common Denominator (LCD)
To eliminate the fractions, we need to combine the two terms on the left side of the equation. This requires a common denominator. The denominators in our equation are (x-1) and (x-2) It's one of those things that adds up. Still holds up..
LCD = (x-1)(x-2)
Step 2: Multiply Both Sides of the Equation by the LCD
This is the crucial step that clears the fractions. We multiply every term in the equation by (x-1)(x-2). This is genuinely important to multiply every term to maintain the equality.
The original equation: x/(x-1) - 1/(x-2) = 0
Multiplying each term by (x-1)(x-2): [(x-1)(x-2)] * [x/(x-1)] - [(x-1)(x-2)] * [1/(x-2)] = 0 * [(x-1)(x-2)]
Now, we simplify by canceling common factors in each term:
- In the first term, (x-1) in the numerator and denominator cancel out, leaving us with: x * (x-2)
- In the second term, (x-2) in the numerator and denominator cancel out, leaving us with: -1 * (x-1)
- The right side of the equation, 0 multiplied by anything, is still: 0
Our equation now looks like this: x(x-2) - 1(x-1) = 0
Step 3: Expand and Simplify the Equation
Next, we use the distributive property to expand the terms But it adds up..
Expand x(x-2): x * x - x * 2 = x² - 2x
Expand -1(x-1): -1 * x - 1 * (-1) = -x + 1
Now, substitute these back into the equation: x² - 2x - x + 1 = 0
Combine the like terms (-2x and -x): x² - 3x + 1 = 0
We have successfully transformed the original rational equation into a standard quadratic equation: x² - 3x + 1 = 0.
Step 4: Solve the Quadratic Equation
A quadratic equation is of the form ax² + bx + c = 0. There are three primary methods to solve such an equation: factoring, completing the square, or using the quadratic formula Worth keeping that in mind. Turns out it matters..
Let's examine our equation: x² - 3x + 1 = 0. Because of that, we attempt to factor it. We need two numbers that multiply to +1 (the product of a and c) and add to -3 (the coefficient b). The factors of 1 are 1 and 1, or -1 and -1.
Since we cannot find two integers that satisfy these conditions, the equation is not factorable using simple integers. So, we will use the quadratic formula, which is a reliable method for any quadratic equation It's one of those things that adds up..
The quadratic formula is: x = [-b ± √(b² - 4ac)] / (2a)
For our equation, x² - 3x + 1 = 0, the coefficients are:
- a = 1
- b = -3
- c = 1
Plugging these values into the formula: x = [-(-3) ± √((-3)² - 4(1)(1))] / (2(1))
Simplify step-by-step: x = [3 ± √(9 - 4)] / 2 x = [3 ± √5] / 2
This gives us two potential solutions: x = (3 + √5) / 2 and x = (3 - √5) / 2
Step 5: Check the Solutions Against the Restrictions
Recall the restrictions we identified at the beginning: x ≠ 1 and x ≠ 2. We must see to it that neither of our potential solutions equals these forbidden values.
- √5 is an irrational number approximately equal to 2.236.
- Let's check x = (3 + √5) / 2 ≈ (3 + 2.236) / 2 ≈ 5.236 / 2 ≈ 2.618. This is not equal to 1 or 2.
- Let's check x = (3 - √5) / 2 ≈ (3 - 2.236) / 2 ≈ 0.764 / 2 ≈ 0.382. This is also not equal to 1 or 2.
Since neither solution violates the domain restrictions, both are valid.
Scientific Explanation: Why This Method Works
The process we followed is grounded in algebraic principles. Consider this: this operation preserves the equality. Worth adding: multiplying by the LCD is a valid operation because we are essentially multiplying the entire equation by the same non-zero expression, (x-1)(x-2). The only risk is introducing "extraneous solutions"—values that satisfy the simplified equation but not the original one Small thing, real impact. That alone is useful..
