Problem 25 - Algebra 1

Question

Simplify the expression. $$\frac{x^{2}-36}{-5 x^{2}} \div(x-6)$$

Step-by-Step Solution

Answer

-\frac{x+6}{5x^{2}}

Step 1: Analyze the Expression

Take a look at the expression \(\frac{x^{2}-36}{-5x^{2}}\) divided by \((x-6)\). The first part of the expression requires simplification in order to resolve the expression, so let's focus on that.

Step 2: Factor the Numerator

The numerator of the fraction, \(x^{2}-36\), is a difference of squares which can be factored into two bins: \(x^{2}-36= (x+6)(x-6)\). So the expression becomes \(\frac{(x+6)(x-6)}{-5x^{2}}\) divided by \((x-6)\).

Step 3: Simplify the Divided Fraction

Now we need to simplify \(\frac{(x+6)(x-6)}{-5x^{2}}\) divided by \((x-6)\). A fraction divided by a number is the same as the fraction multiplied by the reciprocal of that number. Hence, we can rewrite the expression as \(\frac{(x+6)(x-6)}{-5x^{2}}\) times \(\frac{1}{x-6}\).

Step 4: Cancel Common Factors

In this case, \(x-6\) can be seen in both the numerator and the denominator, so they can be canceled out. Therefore, the expression now is \(-\frac{x+6}{5x^{2}}\).

Step 5: Final Simplification

As \(x+6\) and \(5x^{2}\) don't have any common factors, this is the final simplified form of the given expression.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. To achieve this, we need to break down the expression into its most basic parts. This often involves factoring, canceling out similar terms, and performing arithmetic operations like addition and subtraction.

Consider the initial expression \(\frac{x^2 - 36}{-5x^2} \div (x-6)\). To simplify, we need to understand each component and see how they interact with each other. By factoring the above expression, applying arithmetic rules, and eliminating common terms, the expression becomes much easier to handle. Simplification makes expressions more manageable and often reveals hidden relationships between terms.

Factoring

Factoring is the process of breaking down an expression into its components, or factors, that when multiplied together yield the original expression. This is extremely useful, as it often makes simplifying much easier.

In our given problem, the expression \(x^2 - 36\) is clearly a "difference of squares". We can rewrite it as \((x + 6)(x - 6)\) because \(x^2 - 36\) matches the form \(a^2 - b^2 = (a+b)(a-b)\) where \(a = x\) and \(b = 6\).

This step enables us to further simplify the expression, leading to the ultimate goal of the problem - finding the simplest form of the expressions involved.

Fraction Division

Fraction division involves taking the reciprocal of the divisor and then multiplying. This might initially seem complex, but it's quite straightforward once you grasp the idea.

Consider the part of the expression \(\frac{(x+6)(x-6)}{-5x^2} \div (x-6)\). Rather than directly dividing, you take the reciprocal of \((x-6)\), turning the division into a multiplication by \(\frac{1}{x-6}\). Hence, the expression becomes \(\frac{(x+6)(x-6)}{-5x^2} \times \frac{1}{x-6}\).

This simplification is possible because dividing by a number is equivalent to multiplying by its reciprocal - a key concept in fraction division.

Difference of Squares

The "difference of squares" is a specific algebraic pattern where you subtract one squared term from another, leading to expressions like \(a^2 - b^2\). This pattern is pivotal because it can always be factored into \((a+b)(a-b)\).

For example, \(x^2 - 36\) can be viewed as \((x^2 - 6^2)\), a classic difference of squares. It simplifies to \((x+6)(x-6)\). Recognizing this pattern helps transform complex expressions into simpler ones quickly.

By utilizing the difference of squares, we not only simplify but also pave the way for cancelling out terms in expressions like the one in our exercise.