Kent's Equation Transformation Solving For K

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Hey guys! Let's dive into a cool math problem today. We're going to break down how Kent manipulates an equation and what final equation he needs to solve. This is a classic algebra problem that involves clearing fractions and rearranging terms, so stick with me, and we'll conquer it together!

The Initial Equation

Our starting point is this equation:

k + 12/k = 8

Kent's first move is to get rid of that fraction. Fractions can be a bit messy to work with, so a common strategy is to multiply both sides of the equation by the denominator. In this case, the denominator is k. This is a crucial first step in simplifying the equation and making it easier to solve. By eliminating the fraction, we transform the equation into a more manageable form that we can work with directly. Multiplying by k is like using a mathematical magic wand to make the fraction disappear!

The Multiplication Step

So, Kent multiplies both sides of the equation by k. Let's see what happens:

k * (k + 12/k) = 8 * k

We need to distribute that k on the left side. Remember the distributive property? It says that a * (b + c) = a * b + a * c. We're applying that here. This distribution is where the magic truly happens. The k we're multiplying by will interact with both terms inside the parentheses, leading to a simplified equation that's much easier on the eyes. It’s like watching a mathematical puzzle come together, piece by piece.

Distributing and Simplifying

Let's distribute and simplify:

k * k + k * (12/k) = 8k

This simplifies to:

k^2 + 12 = 8k

Notice how the k in the numerator and denominator canceled out in the second term on the left side. This is precisely why we multiplied by k in the first place – to eliminate that pesky fraction! Now, we've got a much cleaner equation to work with, a quadratic equation taking shape before our very eyes. It's a testament to the power of algebraic manipulation!

Now, Kent wants to move all the terms to one side of the equation. This is a standard technique when dealing with quadratic equations. The goal is to set the equation equal to zero. This form allows us to use various methods to solve for k, like factoring, completing the square, or the quadratic formula. It’s like preparing the equation for its final showdown, where we'll uncover the values of k that make it true.

Setting the Stage for Solving

To get all terms on one side, Kent needs to subtract 8k from both sides. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. It’s like a mathematical see-saw; we need to keep both sides level. This step is about bringing order to the equation, arranging the terms in a way that reveals its underlying structure. Think of it as tidying up before the big solve!

The Subtraction Step

Subtracting 8k from both sides, we get:

k^2 + 12 - 8k = 8k - 8k

Which simplifies to:

k^2 + 12 - 8k = 0

We're almost there! Now we have all the terms on one side, and the equation is set equal to zero. But, it looks a little out of order, doesn't it? It’s like having the letters of a word all jumbled up; we need to rearrange them to make sense. The next step is about putting things in the right order, so the equation is in its most recognizable and solvable form.

Standard Form

It's conventional to write quadratic equations in the standard form: ax^2 + bx + c = 0. So, let's rearrange our terms to match this form. It's like putting on our math hats and getting down to business, making sure everything is in its proper place. This standard form is a mathematical blueprint, guiding us toward the solution.

Rearranging into Standard Form

Rearranging the terms, we get:

k^2 - 8k + 12 = 0

This is the quadratic equation Kent needs to solve! We've successfully transformed the original equation into a standard quadratic form. It’s like taking a complicated puzzle and fitting all the pieces together, revealing a clear picture of the equation we need to solve.

So, the equation Kent must solve is:

k^2 - 8k + 12 = 0

This is a quadratic equation, and there are several ways Kent could solve it. He could try factoring, use the quadratic formula, or even complete the square. Each method has its own strengths and weaknesses, and the best choice often depends on the specific equation. It’s like having a toolbox full of different tools; we choose the one that’s best suited for the job.

Factoring: A Possible Route

One common method is factoring. We need to find two numbers that multiply to 12 and add up to -8. Think of it as a mathematical scavenger hunt, searching for the perfect pair of numbers that fit our criteria. Factoring is often the quickest route to the solution, if the equation can be factored easily.

Finding the Factors

The numbers -6 and -2 fit the bill because (-6) * (-2) = 12 and (-6) + (-2) = -8. So, we can factor the equation as follows:

(k - 6)(k - 2) = 0

We've successfully factored the quadratic equation! It’s like unlocking a mathematical secret, revealing the hidden structure of the equation. Now, we're just one step away from finding the solutions.

The Zero Product Property

Now, we use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental principle in algebra, and it's our key to unlocking the solutions. It’s like a mathematical domino effect; if one factor is zero, the whole product crumbles to zero.

Applying the Property

So, either k - 6 = 0 or k - 2 = 0. Solving these simple equations gives us the solutions:

k = 6 or k = 2

We've found the solutions! These are the values of k that make the original equation true. It’s like reaching the end of a mathematical journey, having navigated the twists and turns to arrive at the destination.

The Solutions Unveiled

Therefore, the solutions to the equation are k = 6 and k = 2. We've not only found the equation Kent needed to solve, but we've also gone the extra mile and solved it! It’s a testament to the power of understanding algebraic principles and applying them step-by-step.

In summary, Kent multiplied both sides of the equation by k, rearranged the terms, and ended up with the quadratic equation k^2 - 8k + 12 = 0. We then went a step further and solved this equation, finding the solutions k = 6 and k = 2. This problem illustrates the importance of algebraic manipulation in simplifying and solving equations. It’s like learning a new language; once you understand the grammar and vocabulary, you can express yourself clearly and effectively.

Key Takeaways

  • Multiplying by the denominator clears fractions.
  • Rearranging terms into standard form is crucial for solving quadratic equations.
  • Factoring, the quadratic formula, and completing the square are all methods for solving quadratic equations.
  • The zero product property is a powerful tool for finding solutions once an equation is factored.

By understanding these concepts, you'll be well-equipped to tackle similar algebraic challenges. Keep practicing, and you'll become a math whiz in no time! Remember, math is like a muscle; the more you use it, the stronger it gets. So, keep flexing those mathematical muscles, guys!