Problem 107 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ \frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9} $$

Step-by-Step Solution

Answer

The solutions to the equation \(\frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9}\) are \(x_1 = 3 - 2 \sqrt{2}\) and \(x_2 = 3 + 2 \sqrt{2}\)

Step 1: Simplify the equation

The first step is to simplify the equation. Notice that \(x^{2}-9\) can be factored into \((x - 3)(x + 3)\). This helps to simplify the equation to: \(2x(x + 3) + 6(x - 3) = -28\).

Step 2: Distribute and simplify

Next, distribute the \(x\) on the left side and combine like terms to further simplify the equation. This gives: \(2x^{2} + 3x - 18 = -28\)

Step 3: Solve for x

Now, we can change the equation to the standard form of the quadratic equation: \(2x^{2} + 3x + 10 = 0\). Depress the equation by making substitution \(x=t-{\frac{3}{4}}\) (where \(t\) is a new variable). After that, solve the equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of \(x^{2}\), \(x\), and the constant term.

Step 4: Check your solutions

Finally, it is essential to check the solutions to confirm if they are valid by substituting them back into the original equation and verifying that both sides of the equation are equal. If they are, then the solution is correct.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill for solving quadratic equations effectively. A quadratic equation typically has the form \( ax^2 + bx + c = 0 \). To factor it means to express it as a product of two simpler expressions. In our original exercise, the denominator \( x^2 - 9 \) was simplified by recognizing it as a difference of squares, which factors into \( (x-3)(x+3) \).

• This factoring is helpful because it transforms a complex expression into something easier to work with.
• When factoring, look for common patterns such as the difference of squares, perfect square trinomials, or factor by grouping.

Recognizing these patterns can quickly lead you to a simpler form of the equation, making it easier to solve.

Solving Quadratic Equations

After factoring, the next step is to solve the quadratic equation. Quadratic equations are solved by finding the values of \( x \) that make the equation true. Different methods can be used, such as factoring, completing the square, or using the quadratic formula.

• First, you want the quadratic equation in the form \( ax^2 + bx + c = 0 \).
• Then, choose a solving method appropriate for your equation's complexity.

In our original solution, the equation was manipulated into a more straightforward form before applying a specific method, which illustrates the flexibility in approaching these equations. The key is to isolate \( x \) successfully and solve for its values.

Quadratic Formula

The quadratic formula is a dependable tool that applies to any quadratic equation, whether it can be factored or not. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides a way to find the roots of any quadratic equation where \( a \), \( b \), and \( c \) are the coefficients from \( ax^2 + bx + c = 0 \).

• The discriminant, \( \Delta = b^2 - 4ac \), tells us about the nature of the roots (real or complex).
• Two solutions arise from the "\( \pm \)" sign, reflecting the parabola's intersection points with the x-axis.

It's a universal solution method that guarantees finding all solutions to a quadratic equation, making it a powerful part of your mathematical toolbox.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form, which helps in easily handling equations. Simplifying can involve combining like terms, reducing fractions, or factoring expressions.

• In our exercise, simplifying the expression \( 2x(x+3) + 6(x-3) = -28 \) was necessary to manage the problem effectively.
• Combine like terms: this means adding coefficients of similar terms, reducing complexity.
• A well-simplified expression makes solving and verifying equations more manageable.

By focusing on simplification, we ensure that our steps in solving an equation are clearer and more accurate.