Solving Equations: Transforming K + 12/k = 8
Let's dive into how Kent tackles the equation k + 12/k = 8 and transforms it into a standard quadratic form. Understanding this process is super helpful for anyone learning algebra, so let’s break it down step-by-step. You'll often encounter equations that aren't in the most user-friendly format, and knowing how to manipulate them is a crucial skill. The goal here is to eliminate the fraction and rearrange the terms to get a quadratic equation that's easier to solve. This involves multiplying both sides by a common denominator and then rearranging all terms to one side. By following these steps, Kent converts the original equation into a familiar quadratic form, which he can then solve using various methods, such as factoring, completing the square, or using the quadratic formula. This transformation is a fundamental technique in algebra and is applicable to many different types of equations.
Understanding the Initial Equation
Kent starts with the equation:
k + 12/k = 8
To get rid of the fraction, the first strategic move is to multiply both sides of the equation by k. This is a common technique used to clear denominators in algebraic equations, making them easier to work with. When you multiply each term by k, you eliminate the fraction, which simplifies the equation significantly. This step is crucial because it transforms the equation from one involving a rational expression into a standard polynomial equation. By eliminating the denominator, you can then proceed to rearrange the terms and solve for the variable k. This initial step sets the stage for further simplification and ultimately finding the solution to the equation. Remember, whatever you do to one side of an equation, you must also do to the other side to maintain equality. This principle is fundamental to solving equations in algebra.
Multiplying Through by 'k'
When we multiply both sides by k, we get:
k(k + 12/k) = 8k
Distributing k on the left side:
k * k + k * (12/k) = 8k
This simplifies to:
k^2 + 12 = 8k
This step is a critical part of solving the equation because it eliminates the fraction, making the equation easier to manipulate and solve. By multiplying each term by k, we transform the original equation into a standard polynomial equation. This simplification allows us to rearrange the terms and set the equation equal to zero, which is a necessary step for solving quadratic equations. The distribution of k on the left side ensures that each term is correctly multiplied, maintaining the equality of the equation. This process is a fundamental technique in algebra and is applicable to many different types of equations. It demonstrates the importance of understanding how to manipulate equations to make them more manageable.
Rearranging to Quadratic Form
Now, let's move all the terms to one side to set the equation to zero. This is a standard practice when dealing with quadratic equations because it allows us to use methods like factoring or the quadratic formula to find the solutions. By setting the equation equal to zero, we create a format that is conducive to these solving techniques. This rearrangement is crucial because it transforms the equation into a recognizable form that we can easily work with. The goal is to have all the terms on one side and zero on the other, which simplifies the process of finding the values of the variable that satisfy the equation.
Subtract 8k from both sides:
k^2 + 12 - 8k = 8k - 8k
This gives us:
k^2 - 8k + 12 = 0
Aha! We've successfully transformed the original equation into a quadratic equation in standard form. This form is super useful because it allows us to easily identify the coefficients a, b, and c, which are needed for various solution methods such as the quadratic formula or factoring. Rearranging the equation into this standard form is a fundamental step in solving quadratic equations and provides a clear path towards finding the values of k that satisfy the equation. The process of setting the equation equal to zero simplifies the subsequent steps and allows for a more systematic approach to solving the problem.
The Resulting Quadratic Equation
The equation Kent must now solve is:
k^2 - 8k + 12 = 0
Therefore, the answer is A. k^2 - 8k + 12 = 0
This resulting quadratic equation is in the standard form of ax^2 + bx + c = 0, where a = 1, b = -8, and c = 12. This standard form is essential because it allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. By identifying the coefficients, we can easily substitute them into the appropriate formulas or techniques. This quadratic equation represents the simplified and rearranged form of the original equation, making it much easier to find the values of k that satisfy the condition. The transformation from the original equation to this quadratic form is a key step in solving algebraic problems and demonstrates the importance of manipulating equations to make them more manageable.
Factoring the Quadratic Equation (Optional, but Helpful!)
Just for extra practice, let's factor the quadratic equation k^2 - 8k + 12 = 0. Factoring involves finding two numbers that multiply to 12 (the constant term) and add up to -8 (the coefficient of the k term). These numbers are -6 and -2. This skill of factoring quadratic equations is incredibly valuable because it allows us to quickly find the solutions without resorting to more complex methods. Factoring is often the fastest way to solve quadratic equations when the roots are integers. The process involves identifying the correct factors and then rewriting the equation in a factored form, which can then be easily solved by setting each factor equal to zero.
So we can rewrite the equation as:
(k - 6)(k - 2) = 0
Setting each factor equal to zero gives us:
k - 6 = 0 or k - 2 = 0
Solving for k, we find:
k = 6 or k = 2
These are the solutions to the original equation. Factoring the quadratic equation not only provides the solutions but also confirms that our initial transformation was correct. This process reinforces the importance of understanding how to manipulate algebraic equations and use various techniques to solve them. Factoring is a powerful tool that can simplify complex problems and provide a deeper understanding of the underlying mathematical relationships. This exercise demonstrates the complete process of transforming and solving the equation, from the initial manipulation to finding the final solutions.
Conclusion
So, by multiplying both sides of the original equation k + 12/k = 8 by k and then rearranging the terms, Kent arrives at the quadratic equation k^2 - 8k + 12 = 0. This transformation is a standard technique in algebra used to solve equations involving fractions and is a foundational skill for more advanced mathematical concepts. The ability to manipulate equations and transform them into simpler forms is crucial for problem-solving in various fields. This process not only helps in finding the solutions but also provides a deeper understanding of the underlying mathematical relationships. By understanding these basic principles, students can tackle more complex problems with confidence. The steps involved in this transformation, such as eliminating fractions and rearranging terms, are applicable to a wide range of algebraic problems and are essential for success in mathematics. The resulting quadratic equation is a familiar form that can be easily solved using various methods, making the entire process more manageable.