Problem 123 - College Algebra

Question

List all mumbers that must be excluded from the domain of each rational expression. $$\frac{3}{2 x^{2}+4 x-9}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the rational expression are \(x = -1.86023\) and \(x = 2.36023\).

Step 1: Set Denominator Equal to Zero

The denominator is \(2x^2 + 4x - 9\), set it equal to zero and solve for \(x\). \[2x^2 + 4x - 9 = 0\]

Step 2: Solve for x

This is a quadratic expression and can be solved by factoring, applying the quadratic formula or completing the square. In this case, the quadratic formula is the most suitable method. The quadratic formula is \(x = {-b \pm \sqrt{b^2 - 4ac} \over 2a}\). In this formula, \(a\)=2, \(b\)=4, and \(c\)=-9. Plug these values into the quadratic formula and calculate the value of \(x\). \[x = {-4 \pm \sqrt{4^2 - 4*2*(-9)} \over 2*2}\]

Step 3: Simplify Solution

Simplify the equation to get the final solution for \(x\). After simplifying you will get two solutions which are \(x = -1.86023\) and \(x = 2.36023\).

Key Concepts

Domain of a Function

Understanding the domain of a function is essential when dealing with rational expressions. The domain consists of all possible input values (often "x" values) that will not lead to a mathematical contradiction, such as division by zero. In rational expressions, we must focus on the denominator, as any value that makes the denominator zero must be excluded from the domain.
For example, if we have the expression \(\frac{3}{2x^2 + 4x - 9}\), the domain excludes values of \(x\) for which \(2x^2 + 4x - 9 = 0\). This is why it is crucial to solve this equation to find those specific \(x\) values. Checking what values make the denominator zero ensures that we only consider viable inputs for our function.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, especially when factoring is difficult or impossible. The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
This formula helps find the solutions for any quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants.
In the given problem, we use the quadratic formula with \(a = 2\), \(b = 4\), and \(c = -9\). Substituting these into the formula allows us to calculate the exact values of \(x\) that make the denominator zero, thus identifying the numbers that need to be excluded from the domain.

Factoring

Factoring is one of the primary methods for solving quadratic equations. It involves expressing the quadratic as a product of two simpler expressions. Not all quadratic equations can be easily factored, especially when the solutions are not integers.
To factor a quadratic expression, we often look for two numbers that multiply to \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add to \(b\) (the coefficient of \(x\)).
While factoring is a simpler method for some quadratics, in this problem, due to the nature of the coefficients, the quadratic formula was the more efficient method. Nonetheless, knowing how to factor can still provide insight into some quadratic expressions where other methods become cumbersome.

Completing the Square

Completing the square is another method used to solve quadratic equations. It involves creating a perfect square trinomial from the equation, making it easier to solve by extracting the square root.
For a typical quadratic equation \(ax^2 + bx + c = 0\), the steps to complete the square are as follows:

• Isolate the \(x^2\) and \(x\) terms.
• Divide the coefficient of \(x\) by 2, square it, and add it to both sides of the equation.
• Write the left side as a squared binomial.

Completing the square can be particularly useful for equations that resist easy factoring or when we aim to convert equations to vertex form. In the presented exercise, although the quadratic formula was selected, completing the square provides a valuable alternative, enhancing the understanding and flexibility in solving different quadratic equations.