Add. [tex]$\frac{3}{x}+\frac{x}{x+2}$[/tex]
Answer (2)
To add x 3 and x + 2 x , we find the common denominator, which is x ( x + 2 ) . After rewriting both fractions, we combine the numerators to conclude that the final result is x 2 + 2 x x 2 + 3 x + 6 .
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Find a common denominator: The common denominator for x 3 and x + 2 x is x ( x + 2 ) .
Rewrite each fraction with the common denominator: x 3 = x ( x + 2 ) 3 ( x + 2 ) and x + 2 x = x ( x + 2 ) x 2 .
Add the fractions: x ( x + 2 ) 3 ( x + 2 ) + x ( x + 2 ) x 2 = x ( x + 2 ) 3 x + 6 + x 2 .
Simplify the expression: x 2 + 2 x x 2 + 3 x + 6 .
x 2 + 2 x x 2 + 3 x + 6
Explanation
Understanding the Problem We are asked to add two rational expressions: x 3 and x + 2 x . To do this, we need to find a common denominator and combine the numerators.
Finding the Common Denominator The denominators of the two fractions are x and x + 2 . The least common denominator (LCD) is the product of these two denominators, which is x ( x + 2 ) .
Rewriting Fractions with Common Denominator Now, we rewrite each fraction with the common denominator.
For the first fraction, we multiply the numerator and denominator by ( x + 2 ) : x 3 = x ( x + 2 ) 3 ( x + 2 ) = x ( x + 2 ) 3 x + 6
For the second fraction, we multiply the numerator and denominator by x : x + 2 x = x ( x + 2 ) x ( x ) = x ( x + 2 ) x 2
Adding the Fractions Now that both fractions have the same denominator, we can add them: x ( x + 2 ) 3 x + 6 + x ( x + 2 ) x 2 = x ( x + 2 ) 3 x + 6 + x 2
Simplifying the Expression We can rewrite the numerator in standard form: x ( x + 2 ) x 2 + 3 x + 6
Expanding the denominator gives: x 2 + 2 x x 2 + 3 x + 6
The expression is now simplified.
Final Answer Therefore, the sum of the two rational expressions is x 2 + 2 x x 2 + 3 x + 6 .
Examples
Rational expressions are useful in many areas of science and engineering. For example, in physics, they can be used to describe the relationship between voltage, current, and resistance in an electrical circuit. In chemistry, they can be used to model the rate of a chemical reaction. Understanding how to add and simplify rational expressions is a fundamental skill for solving problems in these fields.
