Problem 54 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{9 p^{2}-30 p+25}{9 p^{2}-25} $$

Step-by-Step Solution

Answer

\(\frac{3p - 5}{3p + 5}\)

Step 1: Identify the form of the expression

Recognize that the given expression is a rational expression which needs to be simplified. The numerator is \(9p^2 - 30p + 25\) and the denominator is \(9p^2 - 25\).

Step 2: Factor the numerator

Factor the quadratic expression in the numerator. Notice that \(9p^2 - 30p + 25\) is a perfect square trinomial: \[9p^2 - 30p + 25 = (3p - 5)^2\].

Step 3: Factor the denominator

Factor the quadratic expression in the denominator. Observe that \(9p^2 - 25\) is a difference of squares: \[9p^2 - 25 = (3p + 5)(3p - 5)\].

Step 4: Simplify the expression

Divide the numerator by the denominator: \[\frac{(3p - 5)^2}{(3p + 5)(3p - 5)}\]. Cancel the common term \(3p - 5\) from the numerator and the denominator: \[\frac{3p - 5}{3p + 5}\].

Key Concepts

Factoring Quadratics

When simplifying rational expressions, the first step is often to factor the quadratic expressions in the numerator and denominator. Begin by identifying common forms like trinomials or binomials. Quadratic expressions follow the general form: \(ax^2 + bx + c\). The process involves finding two numbers that multiply to give \(ac\) and add to give \(b\). This process allows us to rewrite the quadratic in factored form, which can simplify further steps.
For example, consider \(9p^2 - 30p + 25\).

• The product of \(a \cdot c\) is \(9 \cdot 25 = 225\).
• The middle term \(-30p\) breaks down to two roots, \(-15\) and \(-15\).
• This expression factors to \((3p - 5)(3p - 5)\).

Understanding these factorization techniques is crucial for simplifying rational expressions.

Perfect Square Trinomials

A perfect square trinomial is a special quadratic form that equates to the square of a binomial. In simplified terms, it's an expression of the form: \(a^2 \pm 2ab + b^2 = (a \pm b)^2\). Recognizing perfect square trinomials helps you rewrite them quickly and accurately.
Take our example expression, \(9p^2 - 30p + 25\):

• \(9p^2 = (3p)^2\)
• \(-30p = 2 \cdot 3p \cdot (-5)\)
• \(25 = (-5)^2\)

Since the middle term fits, \(9p^2 - 30p + 25\) simplifies to \((3p - 5)^2\). Recognizing these patterns is key to simplifying higher-level algebraic expressions efficiently.

Difference of Squares

The difference of squares is another special quadratic case crucial for simplifying rational expressions. Identified by the form \(a^2 - b^2 = (a + b)(a - b)\), it simplifies two terms squared minus each other. Look at the problem's denominator, \(9p^2 - 25\), which follows this pattern.
Here's how:

• \(9p^2 = (3p)^2\)
• \(25 = (5)^2\)

This reveals our expression to be a difference of squares: \((3p)^2 - (5)^2\). So, \(9p^2 - 25\) factors to \((3p + 5)(3p - 5)\). Mastering the difference of squares factorizations streamlines the problem-solving process.

Simplification

Simplifying rational expressions boils down to canceling common factors after factoring the numerator and the denominator. With our expression now as \(\frac{(3p - 5)^2}{(3p + 5)(3p - 5)}\), we can cancel \((3p - 5)\) present in both. This leaves:
\(\frac{3p - 5}{3p + 5}\).
The aim is to reduce expressions to their simplest form by canceling all possible terms. Always double-check for common factors and validate each step to ensure accuracy. Simplifying expressions makes further algebraic work more manageable and sets the foundation for advanced problem-solving.