Problem 22 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{x}{2 x^{2}+15 x+27} $$

Step-by-Step Solution

Answer

The expression is undefined for \(x = -\frac{9}{2}\) and \(x = -3\).

Step 1: Understand When a Rational Expression is Undefined

A rational expression is undefined when its denominator is equal to zero, as division by zero is not possible.

Step 2: Set Denominator Equal to Zero

To find the numbers for which the rational expression \(\frac{x}{2x^2 + 15x + 27}\) is undefined, set the denominator equal to zero: \(2x^2 + 15x + 27 = 0\).

Step 3: Factor the Quadratic Expression in the Denominator

Factor \(2x^2 + 15x + 27\) to find its roots. We need to find two numbers that multiply to \(2 \times 27 = 54\) and add up to \(15\). These numbers are 9 and 6. Thus, the factoring results in:\[2x^2 + 15x + 27 = (2x + 9)(x + 3).\]

Step 4: Solve for the Values that Make Denominator Zero

Set each factor from the previous step equal to zero and solve for \(x\):1. \(2x + 9 = 0\) leads to \(x = -\frac{9}{2}\).2. \(x + 3 = 0\) leads to \(x = -3\).

Step 5: State the Result

The rational expression \(\frac{x}{2x^2 + 15x + 27}\) is undefined for \(x = -\frac{9}{2}\) and \(x = -3\).

Key Concepts

Understanding Undefined Expressions

A rational expression becomes undefined when its denominator equals zero.
This is because division by zero in mathematics is impossible and has no meaning. In a rational expression like \( \frac{x}{2x^2 + 15x + 27} \), we need to concentrate on the denominator \( 2x^2 + 15x + 27 \).
Finding when this denominator equals zero involves solving this expression:

• Set the denominator equal to zero: \( 2x^2 + 15x + 27 = 0 \).
• The values of \( x \) that satisfy this equation are the points where the expression is undefined.

By doing this, you determine the "problem spots" where the rational expression loses its value.

Factoring Quadratics to Simplify Solving

Factoring is a crucial technique to solve quadratic equations efficiently.
When we need to solve \( 2x^2 + 15x + 27 = 0 \), factoring helps us break the quadratic down into simpler expressions.
To illustrate:

• We look for two numbers that multiply to the product of the coefficient of \( x^2 \) (which is \( 2 \)) times the constant term \( 27 \), so \( 54 \), and simultaneously add up to \( 15 \).
• The numbers 9 and 6 work since \( 9 \times 6 = 54 \) and \( 9 + 6 = 15 \).
• This allows us to factor \( 2x^2 + 15x + 27 \) as \((2x + 9)(x + 3) \).

Factoring simplifies the process, making it straightforward to find points where the quadratic equals zero.

Finding Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
After factoring, solving becomes simpler because you only need to set each factor to zero:

• For \( (2x + 9)(x + 3) = 0 \), individually solve \( 2x + 9 = 0 \) and \( x + 3 = 0 \).
• Solving \( 2x + 9 = 0 \) gives \( x = -\frac{9}{2} \).
• Solving \( x + 3 = 0 \) gives \( x = -3 \).

These values, \( x = -\frac{9}{2} \) and \( x = -3 \), are the solutions to the quadratic. They are precisely where the rational expression \( \frac{x}{2x^2 + 15x + 27} \) becomes undefined as those values make the denominator zero.