Problem 67 - Algebra A Combined Function

Question

Simplify each expression. Then determine whether the given answer is correct. $$ \frac{7-34 x-5 x^{2}}{25 x^{2}-1} ; \text { Answer: } \frac{x+7}{-5 x-1} $$

Step-by-Step Solution

Answer

The given answer \( \frac{x+7}{-5x - 1} \) is correct.

Step 1: Recognize the Quadratic Forms

First, identify that both the numerator and the denominator are quadratic expressions. The numerator is \( -5x^2 - 34x + 7 \) and the denominator is \( 25x^2 - 1 \). Both can potentially be factored.

Step 2: Factor the Denominator

Notice that the denominator \( 25x^2 - 1 \) is a difference of squares. It can be factored as: \[ 25x^2 - 1 = (5x - 1)(5x + 1). \]

Step 3: Attempt to Factor the Numerator

Try factoring the numerator \( -5x^2 - 34x + 7 \). Through trial and simplification, we find that:\[ -5x^2 - 34x + 7 = (-5x - 1)(x + 7). \]

Step 4: Simplify the Expression

With both the numerator and denominator factored, we have:\[ \frac{(-5x - 1)(x + 7)}{(5x - 1)(5x + 1)}. \] Since there are no common factors in the numerator and the denominator, this is the simplest form of the expression.

Step 5: Verify the Given Answer

The given answer is \( \frac{x+7}{-5x - 1} \). Notice that we can write this as \( \frac{-1(x+7)}{-1(-5x - 1)} \), which simplifies to \( \frac{x+7}{-5x - 1} \). Therefore, the provided answer is indeed equivalent to the simplified form we found in Step 4.

Key Concepts

Quadratic Expressions

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic expressions are fundamental because they appear frequently in algebra problems, especially in equations and when working with parabolas.
Understanding the structure of a quadratic expression helps us factor or simplify it, as knowing the coefficients and their positions is critical. A core skill is identifying when an expression is quadratic by its highest degree of 2.

• Example: The expression \( -5x^2 - 34x + 7 \) is quadratic, with \( a = -5 \), \( b = -34 \), and \( c = 7 \).

Quadratic expressions can often be factored into the product of two binomials. This is particularly simple when using special cases, like the difference of squares.

Factoring

Factoring is a method that involves rewriting an expression as the product of simpler expressions, or factors. This is particularly useful in simplifying algebraic expressions and solving equations. The process begins by exploring if the given expression fits a known factoring technique, such as factoring out the greatest common factor, or applying the difference of squares.
To factor quadratic expressions effectively, we typically look for:

• Common factors that can be taken out of each term.
• Special formats like perfect square trinomials or differences of squares.

In our example, the denominator \( 25x^2 - 1 \) is identified as a difference of squares. This allows us to rewrite it as \((5x - 1)(5x + 1)\).
Similarly, regarding the numerator \(-5x^2 - 34x + 7\), trial and error help to discover it can be factored into \((-5x - 1)(x + 7)\).
By recognizing these patterns, we efficiently simplify or solve expressions, uncovering potential simplification opportunities.

Algebraic Fractions

Algebraic fractions are fractions in which the numerator, the denominator, or both, contain algebraic expressions. Simplifying algebraic fractions often involves factoring both the numerator and the denominator, then performing cancellations to reduce the fraction to its simplest form.
To simplify an algebraic fraction completely, follow these steps:

• Factor the numerator and the denominator if possible.
• Cancel any common factors present in both the numerator and the denominator.
• Check the simplified expression for equivalency to ensure no errors were made.

In our example, by factoring both parts of the algebraic fraction, the numerator \((-5x - 1)(x + 7)\) and denominator \((5x - 1)(5x + 1)\) were inspected for common factors. With no common factors to cancel, the expression was confirmed to be in its simplest form. The given answer \( \frac{x+7}{-5x - 1} \) was verified by reshaping the fraction and confirming it is equivalent to the simplified fraction found earlier.
Understanding and practicing these steps helps solidify your ability to manipulate and simplify complex algebraic fractions, making them much more approachable.