Problem 13 - Algebra for College Students

Question

Solve each equation. $$ \frac{2}{2 x-3}-\frac{2}{10 x^{2}-13 x-3}=\frac{x}{5 x+1} $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 6.5\).

Step 1: Identify the Common Denominator

The common denominator involves both sides of the equation; for the terms on the left, it is the product of \((2x-3)\) and \((10x^2 - 13x - 3)\), while the term on the right \((5x+1)\) is also included. Begin by factoring the quadratic \(10x^2 - 13x - 3\).

Step 2: Factor the Quadratic

Factor \(10x^2 - 13x - 3\) into \((5x+1)(2x-3)\) by finding two numbers that multiply to \(-30\) and add to \(-13\). Check the factorization to ensure accuracy.

Step 3: Simplify the Equation Using the Common Denominator

Multiply every term in the equation by the common denominator, \((2x-3)(5x+1)\). This will eliminate the denominators, resulting in a new equation: \(2(5x+1) - 2 = x(2x-3)\).

Step 4: Distribute and Simplify

Expand each side of the equation: \(2 \cdot 5x + 2 - 2 = 10x + 2 = x(2x - 3)\). Simplify further to get \(10x + 2 = 2x^2 - 3x\).

Step 5: Rearrange to Form a Quadratic Equation

Move all terms to one side of the equation: \(0 = 2x^2 - 3x - 10x - 2\). Simplify it to: \(2x^2 - 13x - 2 = 0\).

Step 6: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 2\), \(b = -13\), and \(c = -2\). Calculate \(x = \frac{13 \pm \sqrt{169 + 16}}{4}\) to find the solutions \(x = \frac{13 \pm 13}{4}\).

Step 7: Calculate the Solutions

Solve for \(x\) to get \(x = 0\) or \(x = \frac{26}{4}\). Simplify the solution further to get \(x = 0\) or \(x = 6.5\).

Step 8: Verify the Solutions

Substitute \(x = 0\) and \(x = 6.5\) back into the original equation to ensure they satisfy it. Discard any values that make the denominator zero.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to simplify quadratic equations by expressing them as a product of two binomial expressions. Consider a quadratic expression in the form of \(ax^2 + bx + c\). The goal is to find two numbers that multiply to \(ac\) (the product of the leading coefficient and the constant term) and add up to \(b\) (the coefficient of the \(x\) term). This step is crucial, as it simplifies the equation and makes it easier to solve.

• Example: Take \(10x^2 - 13x - 3\). Here, \(a = 10\), \(b = -13\), and \(c = -3\).
• The product \(ac\) is \(-30\).
• We look for two numbers that multiply to \(-30\) and sum to \(-13\). These numbers are \(-15\) and \(2\).
• Rewrite the expression as \(10x^2 - 15x + 2x - 3\).
• Factor by grouping: \((10x^2 - 15x) + (2x - 3)\) can be grouped and factored to \((5x)(2x - 3) + (1)(2x - 3)\).
• This results in: \((5x + 1)(2x - 3)\).

Factoring simplifies complex expressions, making them manageable and easier to solve.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations that cannot be easily factored. It provides a straightforward way to find the roots of any quadratic equation \[ax^2 + bx + c = 0\].The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use this effectively, you first identify the coefficients \(a\), \(b\), and \(c\) from your equation.

• Step 1: Identify \(a\), \(b\), and \(c\). Let's say for \(2x^2 - 13x - 2\), these values are \(a = 2\), \(b = -13\), and \(c = -2\).
• Step 2: Substitute into the formula: \(x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \times 2 \times (-2)}}{2 \times 2}\).
• Step 3: Calculate inside the square root: \((-13)^2 = 169\) and \(-4 \times 2 \times -2 = 16\). Thus, \(\sqrt{169 + 16} = 13\).
• Step 4: Solve: \(x = \frac{13 \pm 13}{4}\), leading to solutions \(x = 0\) or \(x = 6.5\).

The quadratic formula always works, regardless of whether the quadratic is factorable.

Common Denominator

A common denominator is crucial when solving equations involving fractions. It allows you to combine fractions in a way that eliminates the denominators, making the algebra simpler.Consider the equation: \[\frac{2}{2x-3} - \frac{2}{10x^2-13x-3} = \frac{x}{5x+1}\].To find a common denominator:

• Recognize the different denominators: \((2x-3)\), \((10x^2-13x-3)\), and \((5x+1)\).
• Factor \((10x^2-13x-3)\) as \((5x+1)(2x-3)\) to reveal the common factors.
• The common denominator for the entire equation is \((2x-3)(5x+1)\).
• Multiply every term in the equation by \((2x-3)(5x+1)\) to eliminate all denominators.

Eliminating the denominators simplifies the work, leading to an equation free of fractions, which is much easier to solve.

Equation Solving Steps

Solving algebraic equations involves clear and logical steps to reach the correct solution. These steps help in organizing one's approach, especially when dealing with complex equations.Here is a structured approach:

• Step 1: Identify the problem: Start by understanding the equation and identifying what needs to be done, such as factoring or using the quadratic formula.
• Step 2: Clear fractions: Use the common denominator to remove fractions. Multiply each term by the common denominator as demonstrated in the previous sections.
• Step 3: Simplify: Once fractions are removed, simplify the expressions. Distribute terms and combine like terms effectively.
• Step 4: Reorganize terms: Form a quadratic equation by moving all terms to one side of the equation, leading to the standard form \(ax^2 + bx + c = 0\).
• Step 5: Solve the equation: Depending on the form, factor the equation or use the quadratic formula to find solutions.
• Step 6: Verify: Always check solutions by substituting them back into the original equation to ensure they hold true.

Following these systematic steps ensures you approach each problem with clarity and accuracy.