Math Problem Statement

Select the correct choice below and fill in any answer boxes within your choice. A. StartFraction x squared minus 9 Over x squared EndFraction times StartFraction x squared minus 3 x Over x squared plus 7 x minus 30 EndFraction equals    StartFraction x plus 3 Over x left parenthesis x plus 10 right parenthesis EndFraction ​, xnot equals    0 comma negative 10 comma 3

Solution

We are tasked with simplifying the following expression:

x29x2×x23xx2+7x30\frac{x^2 - 9}{x^2} \times \frac{x^2 - 3x}{x^2 + 7x - 30}

Step 1: Factor the expressions

Let's begin by factoring the numerators and denominators where possible.

  1. Factor x29x^2 - 9: x29=(x3)(x+3)x^2 - 9 = (x - 3)(x + 3)

  2. Factor x23xx^2 - 3x: Factor out an xx: x23x=x(x3)x^2 - 3x = x(x - 3)

  3. Factor x2+7x30x^2 + 7x - 30: Find two numbers that multiply to 30-30 and add up to 77. These numbers are 1010 and 3-3. Thus: x2+7x30=(x3)(x+10)x^2 + 7x - 30 = (x - 3)(x + 10)

Step 2: Substitute the factored forms

Now substitute the factored expressions into the original equation:

(x3)(x+3)x2×x(x3)(x3)(x+10)\frac{(x - 3)(x + 3)}{x^2} \times \frac{x(x - 3)}{(x - 3)(x + 10)}

Step 3: Simplify the expression

Cancel out the common terms:

  • The (x3)(x - 3) terms in the numerator and denominator cancel out.
  • The x2x^2 in the first denominator becomes x×xx \times x, and one xx cancels with the xx in the second numerator.

The simplified expression is now:

x+3x(x+10)\frac{x + 3}{x(x + 10)}

Step 4: Set the domain restrictions

The restrictions on xx come from values that would make any denominator zero. These values are:

  1. From x2x^2: x0x \neq 0
  2. From x3x - 3: x3x \neq 3
  3. From x+10x + 10: x10x \neq -10

Thus, the domain restrictions are x0x \neq 0, x3x \neq 3, and x10x \neq -10.

Final Answer:

x+3x(x+10),x0,10,3\frac{x + 3}{x(x + 10)}, \quad x \neq 0, -10, 3

Would you like further details or have any questions?

Here are five related questions:

  1. How do you factor quadratic expressions?
  2. What are the steps to cancel out terms in a rational expression?
  3. How can you determine domain restrictions from rational expressions?
  4. What happens if a value that makes a denominator zero is included in the solution?
  5. Can factoring help in simplifying complex algebraic fractions?

Tip: Always check for common factors in both the numerator and denominator to simplify expressions easily.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring

Formulas

Difference of Squares: a^2 - b^2 = (a + b)(a - b)
Factoring Quadratic Expressions
Simplifying Rational Expressions

Theorems

Zero-Product Property
Domain Restrictions in Rational Expressions

Suitable Grade Level

Grades 9-10