Math Problem Statement

Which expression is equivalent to the given expression?

Solution

The given expression is:

x26x+9x297\frac{x^2 - 6x + 9}{x^2 - 9} - 7

We need to simplify this expression to find which of the provided options is equivalent. Let's break it down step by step.

  1. Factor the numerator and denominator of the fraction:
    • The numerator x26x+9x^2 - 6x + 9 can be factored as (x3)2(x - 3)^2.
    • The denominator x29x^2 - 9 is a difference of squares and factors as (x3)(x+3)(x - 3)(x + 3).

Thus, the expression becomes:

(x3)2(x3)(x+3)7\frac{(x - 3)^2}{(x - 3)(x + 3)} - 7

  1. Simplify the fraction: Cancel out one factor of (x3)(x - 3) in both the numerator and denominator:

x3x+37\frac{x - 3}{x + 3} - 7

  1. Get a common denominator to combine terms:

[ \frac{x - 3}{x

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring
Simplification

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Factoring theorem
Simplification of rational expressions

Suitable Grade Level

Grades 9-11