Solve F(x) = (7x-19)/(x^2-6x+9): Find A & B

Hey guys! Let's dive into a cool math problem. We're gonna break down a rational function, find some unknown constants, and have a good time doing it. The function we're looking at is $f(x) = \frac{7x - 19}{x^2 - 6x + 9}$. The goal is to rewrite this function in a different form, specifically $f(x) = \frac{A}{x-3} + \frac{B}{(x-3)^2}$, and our mission, should we choose to accept it, is to find the values of A and B. Ready? Let's go!

Understanding the Problem: Partial Fraction Decomposition

So, what's going on here? We're dealing with something called partial fraction decomposition. Basically, it's a technique to break down a complex rational expression (a fraction with polynomials) into simpler fractions. Think of it like taking a complex dish and breaking it down into its individual ingredients. In our case, we have the fraction $(7x - 19) / (x^2 - 6x + 9)$, and we want to rewrite it as the sum of two simpler fractions with unknown numerators (A and B) and denominators based on the factored form of the original denominator. This is a common tactic in calculus, especially when dealing with integration, but it's also a great exercise in algebra and understanding how fractions work.

First, let's look at the denominator of our original function, $x^2 - 6x + 9$. This is a quadratic expression, and we should immediately recognize it as a perfect square trinomial. Factoring it, we get $(x - 3)^2$. This tells us that the decomposition will involve terms with $(x - 3)$ and $(x - 3)^2$ in the denominators, which is exactly what the problem states. This is a crucial step! If you don't recognize the factoring, you'll be stuck right here. Now that we know that $x^2 - 6x + 9 = (x - 3)^2$, we can rewrite our initial equation as $f(x) = \frac{7x - 19}{(x - 3)^2}$.

Our task now is to find the values of A and B such that $\frac{7x - 19}{(x - 3)^2} = \frac{A}{x - 3} + \frac{B}{(x - 3)^2}$.

The Calculation: Finding the Values of A and B

Alright, let's get down to the nitty-gritty and find those elusive values of A and B. The key to solving this is to get rid of those fractions. We can do this by multiplying both sides of the equation by the common denominator, which is $(x - 3)^2$. Multiplying both sides by $(x-3)^2$ gives us:

$$(x - 3)^2 \cdot \frac{7x - 19}{(x - 3)^2} = (x - 3)^2 \cdot \left(\frac{A}{x - 3} + \frac{B}{(x - 3)^2}\right)$$

On the left side, the $(x - 3)^2$ terms cancel out, leaving us with $7x - 19$. On the right side, we distribute the $(x - 3)^2$:

$$7x - 19 = A(x - 3) + B$$

Now, we have a much simpler equation without any fractions! Our next step is to solve for A and B. There are a couple of ways to do this. One method is to expand the right side, collect like terms, and then equate the coefficients of the corresponding powers of x. Alternatively, we can substitute strategic values of x to simplify the equation and solve for A and B. I think that substitution might be the easier way to go.

Let's use the substitution method. First, let's make x = 3. This is a smart choice because it will make the term with A disappear, leaving us with just B. Substituting x = 3:

$$7(3) - 19 = A(3 - 3) + B$$

$$21 - 19 = 0 + B$$

$$2 = B$$

So, we have found that B = 2! Awesome! Now we know B, we can plug this value to find the value of A.

Now that we know B, we can substitute any other value for x. Let's make x = 0

$$7(0) - 19 = A(0 - 3) + 2$$

$$-19 = -3A + 2$$

$$-21 = -3A$$

$$A = 7$$

So, A = 7. Thus, we have found that A = 7 and B = 2! We did it, guys!

Verification: Putting It All Together

Now that we have our values for A and B, let's plug them back into our equation and make sure everything checks out. We should have:

$$\frac{7x - 19}{(x - 3)^2} = \frac{7}{x - 3} + \frac{2}{(x - 3)^2}$$

To verify, let's combine the fractions on the right-hand side using a common denominator, which is $(x - 3)^2$:

$$\frac{7}{x - 3} + \frac{2}{(x - 3)^2} = \frac{7(x - 3)}{(x - 3)^2} + \frac{2}{(x - 3)^2}$$

$$= \frac{7x - 21 + 2}{(x - 3)^2}$$

$$= \frac{7x - 19}{(x - 3)^2}$$

And there it is! The right-hand side simplifies back to our original function. This confirms that our values for A and B are correct. This step is crucial. Always double-check your work to ensure you haven't made any mistakes along the way. It helps reinforce your understanding and gives you confidence in your solution.

Conclusion: We Nailed It!

Alright, folks, we've successfully decomposed the rational function! We started with $f(x) = \frac{7x - 19}{x^2 - 6x + 9}$, and we've rewritten it as $f(x) = \frac{7}{x - 3} + \frac{2}{(x - 3)^2}$. We did this by recognizing the factored form of the denominator, multiplying by the common denominator, and using strategic substitutions to solve for the unknown coefficients A and B. Remember that partial fraction decomposition is a powerful technique. It's used in calculus to solve integration problems. Also, you can apply it to a wide range of other problems involving rational functions. The key is to understand the process and practice, practice, practice! I hope this walkthrough was helpful and that you learned something new. Keep practicing, and you'll become a partial fraction decomposition pro in no time! Keep exploring the world of mathematics, and don't be afraid to tackle new challenges. Math can be fun!

So, to recap, for the function $f(x) = \frac{7x - 19}{x^2 - 6x + 9}$, we have determined that:

• A = 7
• B = 2

That's all, folks! Hope you enjoyed the ride. Until next time, keep those math muscles flexing! Peace out!