Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{4}{2 u-3}+\frac{10}{4 u^{2}-9}=\frac{1}{2 u+3}$$

Step-by-Step Solution

Answer

The solution is \(u = -\frac{25}{6}\).

Step 1: Recognize a common factor

Observe that the denominator of the second and third term can be factored. Notice that the expression \(2u+3\) and \(2u-3\) appear for each denominator separately or as part of a factor.

Step 2: Factor the quadratic expression

The expression \(4u^{2} - 9\) is a difference of squares. It can be factored as \((2u-3)(2u+3)\). The equation now looks like this: \[\frac{4}{2u-3} + \frac{10}{(2u-3)(2u+3)} = \frac{1}{2u+3}.\]

Step 3: Find a common denominator

The common denominator for these fractions will be \((2u-3)(2u+3)\). Multiply each term by the necessary quantity:\[ \frac{4(2u+3)}{(2u-3)(2u+3)} + \frac{10}{(2u-3)(2u+3)} = \frac{(2u-3)}{(2u-3)(2u+3)}.\]

Step 4: Combine the fractions

Combine the fractions under the common denominator:\[ \frac{4(2u+3) + 10}{(2u-3)(2u+3)} = \frac{(2u-3)}{(2u-3)(2u+3)}.\]

Step 5: Simplify the equation

Simplify the numerator of the combined fraction:\[ 4(2u+3) + 10 = 8u + 12 + 10 = 8u + 22. \] Thus,\[ \frac{8u+22}{(2u-3)(2u+3)} = \frac{(2u-3)}{(2u-3)(2u+3)}. \]

Step 6: Equate the numerators

Since the denominators are the same, equate the numerators:\[ 8u + 22 = 2u - 3. \]

Step 7: Solve for the variable "u"

Subtract \(2u\) from both sides:\[ 6u + 22 = -3. \] Then subtract 22 from both sides:\[ 6u = -25. \] Finally, divide by 6:\[ u = -\frac{25}{6}. \]

Step 8: Check the solution validity

Substitute \(u = -\frac{25}{6}\) back into the denominators: - For \(2u-3\) and \(2u+3\): \[2(-\frac{25}{6})-3 = -\frac{50}{6}-\frac{18}{6} = -\frac{68}{6}, \] \[2(-\frac{25}{6})+3 = -\frac{50}{6}+\frac{18}{6} = -\frac{32}{6}. \] Since these values are nonzero, the solution is valid.

Key Concepts

Factoring Quadratics

When solving rational equations, sometimes we come across quadratic expressions that need to be factored. Factoring is the process of rewriting an expression as a product of its linear factors. In our problem, we have the expression \( 4u^2 - 9 \). This is a "difference of squares," a special type of quadratic. A difference of squares takes the form \( a^2 - b^2 \) and factors into \((a-b)(a+b)\).
In our case:

• Identify \(a\) as \(2u\) and \(b\) as 3, since \((2u)^2 = 4u^2\) and \(3^2 = 9\).
• Apply the difference of squares formula to factor \(4u^2 - 9\) into \((2u-3)(2u+3)\).

Factoring is essential because it simplifies the rational equation and allows us to find a common denominator more easily.

Equating Numerators

Once the fractions involved in the rational equation are expressed with a common denominator, we can move on to equating the numerators. This is a critical step because, for two fractions to be equal, both their numerators and denominators must be equal.
In our problem, after we establish a common denominator of \((2u-3)(2u+3)\), the equation becomes:

• \(\frac{8u + 22}{(2u-3)(2u+3)} = \frac{2u-3}{(2u-3)(2u+3)}\).

Since both fractions have an identical denominator, we equate their numerators directly:

• \(8u + 22 = 2u - 3\).

This process of equating numerators is what allows us to solve for the variable \(u\) without worrying about the denominator as long as it is non-zero.

Common Denominator

Finding a common denominator is a fundamental technique in solving rational equations. It allows us to combine and manipulate fractions easily. In this exercise, we identified that the expression \(4u^2 - 9\) could be factored into \((2u-3)(2u+3)\).
To bring each term to a common denominator:

• Multiply the first term by \( (2u+3)/(2u+3) \), resulting in \( \frac{4(2u+3)}{(2u-3)(2u+3)} \).
• No changes are needed for the second term as it already has the common denominator \( (2u-3)(2u+3) \).
• Multiply the third term by \( (2u-3)/(2u-3) \), resulting in \( \frac{(2u-3)}{(2u-3)(2u+3)} \).

By doing this for each term, we can then focus on equating just the numerators, greatly simplifying the problem.

Simplifying Expressions

Simplifying expressions is a key part of solving rational equations efficiently. This involves reducing complex algebraic expressions to their simplest form. In our problem, we begin by simplifying the numerator after achieving a common denominator.
The numerator of the combined fraction:

• Start with \(4(2u+3) + 10\).
• Distribute 4 into \(2u+3\) giving \(8u + 12\).
• Add the resulting expression to 10 to get \(8u + 12 + 10 = 8u + 22\).

Once simplified, this numerator \(8u + 22\) is equated to the numerator from the other side of the equation. Simplifying expressions is not only about making calculations easier but also ensuring that solving the equation for \(u\) becomes straightforward.