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:
Step 1: Factor the expressions
Let's begin by factoring the numerators and denominators where possible.
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Factor :
-
Factor : Factor out an :
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Factor : Find two numbers that multiply to and add up to . These numbers are and . Thus:
Step 2: Substitute the factored forms
Now substitute the factored expressions into the original equation:
Step 3: Simplify the expression
Cancel out the common terms:
- The terms in the numerator and denominator cancel out.
- The in the first denominator becomes , and one cancels with the in the second numerator.
The simplified expression is now:
Step 4: Set the domain restrictions
The restrictions on come from values that would make any denominator zero. These values are:
- From :
- From :
- From :
Thus, the domain restrictions are , , and .
Final Answer:
Would you like further details or have any questions?
Here are five related questions:
- How do you factor quadratic expressions?
- What are the steps to cancel out terms in a rational expression?
- How can you determine domain restrictions from rational expressions?
- What happens if a value that makes a denominator zero is included in the solution?
- 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