Problem 32 - Elementary Algebra

Question

\(\frac{x}{x-4}-2=\frac{4}{x-4}\)

Step-by-Step Solution

Answer

The solutions are \( x = 3 + \sqrt{5} \) and \( x = 3 - \sqrt{5} \).

Step 1: Analyze the Equation

The given equation is \( \frac{x}{x-4} - 2 = \frac{4}{x-4} \). Notice that both fractions have a common denominator of \( x-4 \). We're going to eliminate the fractions by multiplying through by this common denominator.

Step 2: Eliminate Fractions

Multiply every term of the equation by \( x-4 \) to eliminate the denominators: \[(x)(x-4) - 2(x-4) = 4\]Simplify each term: \[x^2 - 4x - 2x + 8 = 4\]Combine like terms: \[x^2 - 6x + 8 = 4\]

Step 3: Solve the Quadratic Equation

Subtract 4 from both sides to set the equation to zero:\[x^2 - 6x + 4 = 0\]This is a quadratic equation in standard form. We can solve it using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For this equation, \( a = 1 \), \( b = -6 \), and \( c = 4 \).

Step 4: Apply the Quadratic Formula

Plug the values into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 1 \times 4}}{2 \times 1} \]Simplifying inside the square root:\[ x = \frac{6 \pm \sqrt{36 - 16}}{2} \]\[ x = \frac{6 \pm \sqrt{20}}{2} \]Further simplifying the square root and fraction:\[ x = \frac{6 \pm 2\sqrt{5}}{2} \]\[ x = 3 \pm \sqrt{5} \]

Step 5: Verify Solutions and Consider Restrictions

Check both solutions in the context of the original equation:- \( x = 3 + \sqrt{5} \) and \( x = 3 - \sqrt{5} \).Since neither solution equals 4, both solutions are valid for the original equation's denominator \( x-4 \) not being zero.

Key Concepts

Step-by-step algebra

When tackling algebra problems, particularly equations, breaking them down into manageable steps is crucial. Every equation hides a path to its solution. The key lies in unraveling it step-by-step. Here’s how you can approach solving problems systematically:

• Start by identifying key components: look at the structure and simplify where possible.
• Next, analyze what methods are needed: substitution, elimination, or factoring might be necessary.
• Execute one step at a time: keep it simple, solve small chunks, and always double-check after each step.

By dissecting a problem like our example, we expose its "skeleton" - such as the quadratic in our solution. Breaking down these steps makes following even complex algebra accessible.

Fractions in equations

Fractions can make equations look challenging but are not impossible to work with. In equations, fractions often appear as ratios or to express parts of a quantity. Here’s how you can tackle them:

• Identify the common denominator: this allows you to combine fractions seamlessly.
• Eliminate fractions by multiplying through by the least common denominator (LCD), as seen in the example provided.
• Simplify fractions early: if terms can be cancelled or factored, do this first to make calculations easier.

By transforming an equation with fractions into one without, it eases the path to solution. Practice simplicity in handling numerators and denominators, and soon equations with fractions will not seem daunting.

Quadratic formula application

The quadratic formula is a powerful tool to solve any quadratic equation, even when other methods like factoring are not feasible. To use the quadratic formula effectively, here are the steps to follow:

• Set your quadratic equation to standard form: where the equation looks like \(ax^2 + bx + c = 0\).
• Identify coefficients \(a\), \(b\), and \(c\): from the standard form.
• Substitute these into the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• Simplify step-by-step: calculate inside the square root first, then apply additions and subtractions.

When using the quadratic formula, pay special attention to the discriminant \(b^2 - 4ac\). It indicates the nature of the roots: real or complex. Mastery of this formula equips you with a means to tackle any quadratic problem efficiently.