Simplify $\frac{9 x^2+12}{6}$
Answer (2)
Find the greatest common divisor (GCD) of the coefficients in the numerator and the denominator.
Factor out the GCD from the numerator: 9 x 2 + 12 = 3 ( 3 x 2 + 4 ) .
Divide both the numerator and the denominator by the GCD: 6 3 ( 3 x 2 + 4 ) = 2 3 x 2 + 4 .
The simplified expression is 2 3 x 2 + 4 .
Explanation
Understanding the problem We are asked to simplify the expression 6 9 x 2 + 12 . To do this, we will first find the greatest common divisor (GCD) of the coefficients in the numerator and the denominator.
Finding the GCD The numerator is 9 x 2 + 12 . The coefficients are 9 and 12. The denominator is 6. The GCD of 9, 12, and 6 is 3.
Factoring out the GCD We can factor out the GCD, 3, from the numerator: 9 x 2 + 12 = 3 ( 3 x 2 + 4 ) . So the expression becomes 6 3 ( 3 x 2 + 4 ) .
Simplifying the fraction Now we can simplify the fraction by dividing both the numerator and the denominator by 3: 6 3 ( 3 x 2 + 4 ) = 2 3 x 2 + 4 .
Final Answer Therefore, the simplified expression is 2 3 x 2 + 4 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in many real-world applications. For example, when designing a bridge, engineers use rational expressions to model the forces and stresses acting on the structure. By simplifying these expressions, they can more easily analyze the behavior of the bridge and ensure its stability. Similarly, in economics, rational expressions are used to model supply and demand curves. Simplifying these expressions can help economists understand the relationship between price and quantity and make predictions about market behavior.
To simplify 6 9 x 2 + 12 , first find the GCD of the coefficients, which is 3. Factor the numerator to get 6 3 ( 3 x 2 + 4 ) and then simplify it to obtain 2 3 x 2 + 4 . The final answer is 2 3 x 2 + 4 .
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