Simplify: [tex]$\frac{a^2}{a-3}-\frac{9-6 a}{3-a}$[/tex]
Answer (1)
Rewrite the second fraction with denominator a − 3 : a − 3 a 2 − − ( a − 3 ) 9 − 6 a = a − 3 a 2 + a − 3 9 − 6 a .
Combine the fractions over the common denominator: a − 3 a 2 + 9 − 6 a .
Rewrite the numerator as a perfect square: a − 3 a 2 − 6 a + 9 = a − 3 ( a − 3 ) 2 .
Cancel the common factor: a − 3 ( a − 3 ) 2 = a − 3 . The simplified expression is a − 3 .
Explanation
Understanding the Problem We are asked to simplify the expression a − 3 a 2 − 3 − a 9 − 6 a .
Rewriting the Expression Notice that the denominators a − 3 and 3 − a are negatives of each other. We can rewrite the second fraction with the denominator a − 3 by factoring out a − 1 from the denominator: a − 3 a 2 − 3 − a 9 − 6 a = a − 3 a 2 − − ( a − 3 ) 9 − 6 a = a − 3 a 2 + a − 3 9 − 6 a .
Combining Fractions Now that the denominators are the same, we can combine the fractions over the common denominator: a − 3 a 2 + 9 − 6 a .
Simplifying the Numerator We can rewrite the numerator as a perfect square: a − 3 a 2 − 6 a + 9 = a − 3 ( a − 3 ) 2 .
Canceling Common Factors Finally, we can cancel the common factor of a − 3 from the numerator and denominator, assuming a = 3 : a − 3 ( a − 3 ) 2 = a − 3.
Final Answer Therefore, the simplified expression is a − 3 .
Examples
Understanding how to simplify algebraic expressions like this is crucial in many areas, such as physics and engineering. For example, when calculating the trajectory of a projectile, you might encounter complex expressions that need simplification to find the range or maximum height. Similarly, in electrical engineering, simplifying circuit equations often involves algebraic manipulation to determine current and voltage relationships. These skills are also valuable in computer science, particularly in algorithm design and optimization, where simplifying expressions can lead to more efficient code.
