Simplify (9x-36)/(x^2+7x-18)

The question is asking to perform algebraic simplification on the given rational expression. It involves reducing the expression (9x-36)/(x^2+7x-18) to its simplest form by factoring the numerator and the denominator and then canceling out any common factors that appear in both. This is typically done by factorizing the quadratic equation in the denominator and finding common factors with the linear expression in the numerator that can be divided out.

$\frac{9 x - 36}{x^{2} + 7 x - 18}$

Answer

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Solution:

Step 1: Extract the common factor from the numerator

Extract the factor of $9$ from the terms in the numerator.

Step 1.1: Factor out $9$ from $9 x$

Rewrite $9 x - 36$ by factoring out $9$ : $\frac{9 (x) - 36}{x^{2} + 7 x - 18}$

Step 1.2: Factor out $9$ from $- 36$

Continue factoring: $\frac{9 x + 9 \cdot (- 4)}{x^{2} + 7 x - 18}$

Step 1.3: Combine the factored terms

Combine the factored terms in the numerator: $\frac{9 (x - 4)}{x^{2} + 7 x - 18}$

Step 2: Factor the denominator using the AC method

Factor the quadratic expression in the denominator.

Step 2.1: Find two numbers with a product of $c$ and a sum of $b$

Identify two integers whose product equals $- 18$ and whose sum equals $7$ : $- 2$ and $9$

Step 2.2: Write the denominator in factored form

Express the denominator in its factored form using the identified integers: $\frac{9 (x - 4)}{(x - 2) (x + 9)}$

Knowledge Notes:

The problem involves simplifying a rational expression. The process requires factoring both the numerator and the denominator. Here are the relevant knowledge points:

Factoring out common factors: This is the process of identifying and extracting common factors from terms in an expression. For example, $9 x - 36$ can be factored as $9 (x - 4)$ because both terms share a common factor of $9$ .
Quadratic factoring (AC method): This method is used to factor quadratic expressions of the form $a x^{2} + b x + c$ . The AC method involves finding two numbers that multiply to $a c$ and add up to $b$ . These two numbers are then used to split the middle term and factor by grouping.
Simplifying rational expressions: Once the numerator and denominator are factored, any common factors can be canceled out to simplify the expression. In this case, if there were common factors in the numerator and denominator, they would be canceled. However, in the given problem, there are no common factors to cancel after factoring.
Latex formatting: Expressions are rendered in Latex format to clearly display mathematical operations and structure. For example, $\frac{9 (x - 4)}{(x - 2) (x + 9)}$ is a Latex-rendered expression showing the simplified form of the rational expression.