Solving Quadratic Equations K Squared 8k 12

Solving equations is a fundamental skill in mathematics, and understanding the steps involved in transforming an equation is crucial for finding the solution. In this article, we will explore a specific equation transformation scenario and delve into the underlying mathematical principles. We'll break down the process, discuss the importance of each step, and highlight the significance of understanding the resulting quadratic equation. This deep dive will provide a comprehensive understanding of how to tackle such problems effectively. Let's embark on this mathematical journey to fully grasp the nuances of equation manipulation and the powerful techniques used to solve quadratic equations.

The Initial Equation

Kent starts with the equation:

$$k + \frac{12}{k} = 8$$

This equation involves a variable k and a fraction with k in the denominator. To solve this equation, Kent decides to multiply both sides by an expression involving k. This is a common strategy to eliminate the fraction and simplify the equation. The key here is to understand why this manipulation is valid and how it transforms the original equation into a more manageable form. The initial equation presents a challenge because of the fractional term, and Kent's approach aims to create a more standard algebraic form that can be solved using familiar techniques. Let's explore the reasons behind this approach and the subsequent steps involved in detail.

Multiplying by k

The expression Kent uses to multiply both sides is k. This is a crucial step in eliminating the fraction. Multiplying both sides of the equation by k gives:

$$k\left(k + \frac{12}{k}\right) = 8k$$

Distributing k on the left side, we get:

$$k^2 + 12 = 8k$$

This step is pivotal because it transforms the original equation, which involved a fraction, into a polynomial equation. The multiplication ensures that the denominator k in the fractional term is canceled out, leading to a simpler algebraic expression. This transformation is based on the fundamental principle that multiplying both sides of an equation by the same non-zero quantity preserves the equality. By doing so, Kent is setting the stage for solving the equation using techniques applicable to polynomial equations. It's essential to recognize the significance of this step in simplifying the equation and preparing it for further manipulation.

Rearranging the Terms

Kent then moves all the terms to one side of the equation. This is a standard technique for solving quadratic equations. Subtracting 8k from both sides, we get:

$$k^2 - 8k + 12 = 0$$

This resulting equation is a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = -8, and c = 12. Recognizing 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 rearranging the terms and setting the equation equal to zero, Kent has prepared the equation for these solution techniques. The significance of this step lies in transforming the equation into a form that is amenable to well-established methods for finding the values of k that satisfy the equation. Understanding the standard form of a quadratic equation is crucial for selecting the appropriate solution method.

The Quadratic Equation to Solve

The equation Kent must solve now is:

$$k^2 - 8k + 12 = 0$$

This quadratic equation is the result of Kent's transformations. To find the solutions for k, we can use factoring, completing the square, or the quadratic formula. Let's explore each of these methods in the context of this equation.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. We look for two numbers that multiply to 12 (the constant term) and add up to -8 (the coefficient of the k term). These numbers are -2 and -6. Thus, we can factor the equation as:

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

Setting each factor equal to zero gives the solutions k = 2 and k = 6. Factoring is a powerful technique when applicable, as it provides a direct route to the solutions. However, not all quadratic equations can be easily factored, making it necessary to consider alternative methods.

Completing the Square

Completing the square involves transforming the quadratic equation into a perfect square trinomial. This method can be used for any quadratic equation. To complete the square, we take half of the coefficient of the k term (-8), square it ((-4)² = 16), and add and subtract it within the equation:

$$k^2 - 8k + 16 - 16 + 12 = 0$$

This gives:

$$(k - 4)^2 - 4 = 0$$

Rearranging the terms, we get:

$$(k - 4)^2 = 4$$

Taking the square root of both sides:

$$k - 4 = \pm 2$$

Thus, k = 4 + 2 = 6 and k = 4 - 2 = 2, which are the same solutions obtained by factoring. Completing the square is a versatile method that works for all quadratic equations, and it forms the basis for deriving the quadratic formula.

Quadratic Formula

The quadratic formula is a general solution for quadratic equations of the form ax² + bx + c = 0. The formula is:

$$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

For our equation, a = 1, b = -8, and c = 12. Plugging these values into the formula:

$$k = \frac{8 \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$$

$$k = \frac{8 \pm \sqrt{64 - 48}}{2}$$

$$k = \frac{8 \pm \sqrt{16}}{2}$$

$$k = \frac{8 \pm 4}{2}$$

This gives k = (8 + 4)/2 = 6 and k = (8 - 4)/2 = 2, consistent with the solutions obtained by factoring and completing the square. The quadratic formula is a powerful tool that provides a guaranteed solution for any quadratic equation, making it an essential technique in algebra.

Importance of the Transformation

The transformation Kent performed is essential for solving the original equation because it eliminates the fraction and converts the equation into a standard quadratic form. This allows us to apply well-known techniques for solving quadratic equations. The ability to manipulate equations and transform them into solvable forms is a critical skill in mathematics. Understanding these transformations not only helps in solving equations but also provides a deeper understanding of the underlying mathematical principles. The process Kent followed demonstrates a systematic approach to problem-solving, emphasizing the importance of simplification and standardization in mathematical manipulations. This approach is applicable to a wide range of mathematical problems, highlighting the versatility of these techniques.

Checking the Solutions

It is always a good practice to check the solutions in the original equation. For k = 2:

$$2 + \frac{12}{2} = 2 + 6 = 8$$

For k = 6:

$$6 + \frac{12}{6} = 6 + 2 = 8$$

Both solutions satisfy the original equation, confirming their validity. Checking the solutions is a crucial step in the problem-solving process, as it ensures that the values obtained are indeed the correct answers. This step helps to catch any errors made during the solution process and reinforces the understanding of the equation's properties. By verifying the solutions, we gain confidence in the accuracy of our work and the correctness of the approach used.

Conclusion

In this article, we examined the steps Kent took to transform and solve an equation. By multiplying both sides by k and rearranging the terms, Kent arrived at the quadratic equation k² - 8k + 12 = 0. We then explored three methods for solving this quadratic equation: factoring, completing the square, and the quadratic formula. Each method led to the same solutions, k = 2 and k = 6. This exercise demonstrates the importance of algebraic manipulation and the versatility of different methods for solving quadratic equations. The ability to transform equations and choose the appropriate solution technique is a fundamental skill in mathematics, and understanding these concepts is crucial for success in algebra and beyond. The journey from the initial equation to the final solutions highlights the power of mathematical reasoning and the elegance of the techniques used to solve these problems.