Problem 21 - Precalculus

Question

Solve each equation. $$\frac{5}{x-4}-\frac{3}{2 x^{2}-5 x-12}=\frac{1}{2 x+3}$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{16}{9}\).

Step 1: Factor the Quadratic

Let's start by factoring the quadratic in the denominator of the second fraction, which is \(2x^2 - 5x - 12\). We need to find two numbers that multiply to \(-24\) (the product of \(2\) and \(-12\)) and add to \(-5\). These numbers are \(3\) and \(-8\). Thus, we can rewrite the quadratic as \(2x^2 - 8x + 3x - 12\).

Step 2: Group and Factor by Grouping

From the expression \(2x^2 - 8x + 3x - 12\), group into pairs and factor: \((2x^2 - 8x) + (3x - 12)\). From the first group, factor out \(2x\) to get \(2x(x - 4)\). From the second group, factor out \(3\) to get \(3(x - 4)\). Thus, the quadratic factors as \((2x + 3)(x - 4)\).

Step 3: Establish a Common Denominator

The common denominator for the equation is \((x - 4)(2x + 3)\). Multiply each term by this common denominator to clear the fractions. For \(\frac{5}{x-4}\), multiply by \((2x + 3)\). For the term \(-\frac{3}{2x^2 - 5x - 12}\), it is already in this form. Finally, for \(\frac{1}{2x + 3}\), multiply by \(x - 4\).

Step 4: Simplify and Solve the Equation

After multiplying by the common denominator, the equation becomes \(5(2x + 3) - 3 = (x - 4)\). Expand and simplify: \(10x + 15 - 3 = x - 4\), which simplifies to \(10x + 12 = x - 4\).

Step 5: Isolate the Variable

Rearrange the equation to isolate \(x\) on one side: \(10x - x = -4 - 12\), leading to \(9x = -16\). Divide both sides by \(9\) to solve for \(x\), yielding \(x = -\frac{16}{9}\).

Step 6: Verify the Solution

Substitute \(x = -\frac{16}{9}\) back into the original equation to ensure there are no invalid operations such as division by zero. The denominator terms are non-zero for this value of \(x\), confirming it's a valid solution.

Key Concepts

Factoring Quadratic Expressions

When working with quadratic equations, one of the key steps is to factor them. This process involves breaking down a quadratic equation into simpler expressions that can be multiplied to get the original equation. It typically looks like this: if you have the quadratic expression
\(ax^2 + bx + c\), you need to find two numbers that:

• Multiply to give the product of \(a\) and \(c\)
• Add up to \(b\)

To demonstrate, let's consider the quadratic expression \(2x^2 - 5x - 12\). We look for two numbers that multiply to \(-24\) (\(2 \times -12\)) and add to \(-5\). Here, the numbers are \(3\) and \(-8\). Thus, we can rewrite our expression as \(2x^2 - 8x + 3x - 12\). This method of breaking down a quadratic equation is crucial because it simplifies further operations. It makes finding solutions less complex, especially in the context of solving rational equations.

Common Denominator

A common denominator is essential when solving equations that involve fractions. It’s the least common multiple of the denominators that allows you to combine the fractions into one expression without changing their size.
In the given exercise, we have fractions with different denominators: \(x - 4\), \(2x^2 - 5x - 12\) which factors into \((2x + 3)(x - 4)\), and \(2x + 3\).
To solve the equation, establish a common denominator, which in this case is \((x - 4)(2x + 3)\). Once you have a common denominator, each term in the equation can be manipulated or combined. This makes the equation easier to solve as the fractions are cleared after multiplying the entire equation by this common denominator. Multiplying each term by the common denominator essentially "removes" the denominators, leaving you with an equation that is simpler to handle.

Solve Rational Equations

Solving rational equations involves finding unknown variables in equations that have rational expressions or fractions. In these equations, the variable appears in the denominator, which makes finding solutions a bit tricky.
For the exercise problem, after identifying a common denominator and multiplying through to clear the fractions, you reach a simpler algebraic equation:
\(5(2x + 3) - 3 = (x - 4)\).
This expression expands and simplifies to \(10x + 12 = x - 4\). This step turns a complex rational equation into a linear one, making it more straightforward to solve. The key is always to eliminate the fractions early on, using methods like factoring and finding common denominators, as these simplify the entire problem-solving process.

Variable Isolation in Equations

Once you have simplified an equation by eliminating fractions, the next step is to isolate the variable. This means rearranging the equation so that the variable, often \(x\), stands alone on one side of the equation.
From the step where we have \(10x + 12 = x - 4\), our goal is to get \(x\) by itself. To isolate \(x\), subtract \(x\) from both sides: \(10x - x = -4 - 12\).
This simplifies to \(9x = -16\).

• Next, divide both sides by \(9\) to solve for \(x\): \(x = -\frac{16}{9}\).

The process of isolating the variable involves using simple algebraic operations: addition, subtraction, multiplication, or division. By doing this, you solve for the variable, revealing its value in the context of the equation. Remember to always check your solution by substituting it back into the original equation to ensure it doesn’t lead to any impossible operations such as division by zero.