Problem 45 - Algebra 2

Question

Simplify each expression. \(\frac{b^{2}-4 b+3}{3-2 b-b^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( -\frac{b-3}{b+3} \).

Step 1: Recognize Quadratics

Notice that both the numerator and denominator of the fraction are quadratic expressions. The numerator is given by \(b^2 - 4b + 3\) and the denominator is \(3 - 2b - b^2\).

Step 2: Rearrange the Denominator

Rearrange the denominator for simplicity by writing it in standard form: \(3 - 2b - b^2 = -b^2 - 2b + 3\). This can be further rearranged as \(- (b^2 + 2b - 3)\).

Step 3: Factor the Numerator

Factor the quadratic expression in the numerator: \(b^2 - 4b + 3\). This can be factored as \((b - 3)(b - 1)\).

Step 4: Factor the Denominator

Factor the quadratic expression in the rearranged denominator: \(- (b^2 + 2b - 3)\). First, factor \(b^2 + 2b - 3\) as \((b + 3)(b - 1)\). Thus, \(- (b + 3)(b - 1)\).

Step 5: Simplify the Expression

Now that we have the fraction \( \frac{(b - 3)(b - 1)}{-(b + 3)(b - 1)} \), notice that \((b - 1)\) is common in both the numerator and the denominator, so they cancel each other out. The expression simplifies to \(- \frac{b - 3}{b + 3} \).

Step 6: Final Simplified Form

The final simplified form of the expression is \(- \frac{b - 3}{b + 3}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a key step in simplifying rational expressions. A quadratic expression typically takes the form \( ax^2 + bx + c \). Our mission here is to express it as a product of two binomials. Typically, you'll look to express a quadratic like \( b^2 - 4b + 3 \) into the form \((b - m)(b - n)\).
To achieve this, you begin by identifying pairs of numbers whose product yields the constant term \( c \) and whose sum provides the linear coefficient \( b \). For example, in \( b^2 - 4b + 3 \):

• The product should be \( +3 \).
• The sum should be \( -4 \).

The numbers satisfying these conditions are \(-1\) and \(-3\). Thus, the expression factors to \((b - 3)(b - 1)\).
Finding the correct pair might require a little trial and error, but this becomes faster with practice.

Cancelling Common Factors

Once the expressions in the numerator and denominator are factored, the next step is to identify any common factors. Cancelling out common factors is similar to reducing fractions where you divide both the top and the bottom by the same number.
In our exercise, after factoring, you're left with:

• Numerator: \((b - 3)(b - 1)\)
• Denominator: \(-(b + 3)(b - 1)\)

You can see \((b - 1)\) is present in both the numerator and the denominator. By cancelling these out, you're simplifying the expression effectively. It's important to remember to handle any negative signs carefully, as they can affect the overall expression if not managed properly.
After cancelling, our expression simplifies to \(- \frac{b - 3}{b + 3}\). This shows how eliminating common factors makes the expression simpler.

Rearranging Expressions

Rearranging expressions is sometimes necessary to make them easier to factor. It involves rewriting the expression into a standard form, allowing the structure to be more apparent.
For quadratics, this means reshuffling parts of the expression to align it into \( ax^2 + bx + c \). In our case, the denominator started off as \( 3 - 2b - b^2 \).
Rearranging the terms, it becomes \( -b^2 - 2b + 3 \), or, by factoring out a negative sign, \(- (b^2 + 2b - 3)\). This factoring out helps us directly see the expression needed for simple factoring.
Rearranging is a powerful tool, allowing you to reveal the true, often hidden, structure of an expression and making subsequent steps like factoring straightforward. It might seem like a minor shuffle, but it is essential for a successful simplification process.