Solving Quadratic Equations Completing The Square For X^2-16x+64=-8

In the realm of mathematics, quadratic equations hold a significant place, appearing in various applications ranging from physics to engineering. One of the fundamental methods for solving quadratic equations is completing the square. This technique transforms a quadratic equation into a perfect square trinomial, making it easier to find the roots. This article provides an in-depth exploration of completing the square, focusing on the equation $x^2 - 16x + 64 = -8$ and its transformation into the form $(x - \square)^2 = -8$.

Understanding Quadratic Equations

Before delving into the specifics of completing the square, it's crucial to grasp the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are called its roots or zeros, representing the values of x that satisfy the equation. These roots can be real or complex numbers.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations, each with its advantages and applications:

• Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It's effective for equations with easily factorable expressions.
• Quadratic Formula: A universal formula that provides the solutions for any quadratic equation, regardless of its factorability. The quadratic formula is given by $x = (-b ± √(b^2 - 4ac)) / (2a)$.
• Completing the Square: A technique that transforms the quadratic equation into a perfect square trinomial, making it easier to isolate the variable x. This method is particularly useful for understanding the structure of quadratic equations and deriving the quadratic formula.

Completing the Square: A Step-by-Step Approach

Completing the square is a powerful technique for rewriting a quadratic expression in a form that reveals its key properties. The goal is to transform the expression into a perfect square trinomial, which can be factored as $(x + k)^2$ or $(x - k)^2$, where k is a constant. Let's break down the process step by step.

1. Understanding Perfect Square Trinomials

At the heart of completing the square lies the concept of a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. For instance, $x^2 + 2kx + k^2$ is a perfect square trinomial because it can be factored as $(x + k)^2$. Similarly, $x^2 - 2kx + k^2$ can be factored as $(x - k)^2$. Recognizing this pattern is essential for completing the square.

2. Manipulating the Quadratic Equation

The process of completing the square involves manipulating the quadratic equation to create a perfect square trinomial. This typically involves the following steps:

1. Isolate the Constant Term: Move the constant term (c) to the right side of the equation.
2. Divide by the Leading Coefficient (if necessary): If the coefficient of the $x^2$ term (a) is not 1, divide both sides of the equation by a. This ensures that the coefficient of $x^2$ is 1, which is necessary for completing the square.
3. Complete the Square: Take half of the coefficient of the x term (b), square it, and add it to both sides of the equation. This step is crucial for creating the perfect square trinomial. The value added to both sides is $(\frac{b}{2})^2$.
4. Factor the Perfect Square Trinomial: Factor the left side of the equation as a squared binomial. This should be in the form $(x + k)^2$ or $(x - k)^2$.
5. Solve for x: Take the square root of both sides of the equation, remembering to consider both positive and negative roots. Then, solve for x.

Applying Completing the Square to $x^2 - 16x + 64 = -8$

Now, let's apply the completing the square technique to the given equation, $x^2 - 16x + 64 = -8$. This example is particularly interesting because the left-hand side is already a perfect square trinomial. However, it provides a good illustration of the process and how to represent the equation in the desired form.

1. Recognizing the Perfect Square Trinomial

The expression $x^2 - 16x + 64$ is a perfect square trinomial. To see this, we can observe that 64 is the square of 8 ($8^2 = 64$), and -16 is -2 times 8 (-2 * 8 = -16). Thus, the expression can be factored as $(x - 8)^2$.

2. Rewriting the Equation

Since $x^2 - 16x + 64$ is already a perfect square trinomial, we can directly rewrite the equation as:

$(x - 8)^2 = -8$

This is the desired form, $(x - \square)^2 = -8$, where the square is 8. The equation is now in a form that readily shows the vertex form of the parabola represented by the quadratic expression. This form is particularly useful for graphing the quadratic function and identifying its key features.

3. Solving for x (Optional)

While the question primarily focuses on rewriting the equation, let's briefly discuss solving for x. To solve for x, we take the square root of both sides:

$√(x - 8)^2 = ±√(-8)$

$x - 8 = ±√(8)i$

$x = 8 ± 2√2i$

Here, we encounter complex roots because the square root of a negative number involves the imaginary unit i. This demonstrates that the equation has no real solutions, which is also evident from the fact that a squared term cannot be negative in the realm of real numbers.

Benefits of Completing the Square

Completing the square offers several advantages beyond just solving quadratic equations:

• Understanding Quadratic Forms: It provides a deeper understanding of the structure of quadratic expressions and their relationship to perfect square trinomials.
• Deriving the Quadratic Formula: The quadratic formula itself can be derived by completing the square on the general quadratic equation $ax^2 + bx + c = 0$. This highlights the fundamental nature of the technique.
• Vertex Form of a Parabola: Completing the square allows us to rewrite a quadratic equation in vertex form, $y = a(x - h)^2 + k$, where (h, k) represents the vertex of the parabola. This form is invaluable for graphing quadratic functions and identifying their maximum or minimum values.
• Solving Optimization Problems: In various optimization problems, completing the square can help identify the maximum or minimum values of quadratic functions, which is crucial in fields like engineering and economics.

Conclusion

Completing the square is a fundamental technique in mathematics with applications extending beyond solving quadratic equations. It provides valuable insights into the structure of quadratic expressions, helps in deriving the quadratic formula, and facilitates the transformation of quadratic equations into vertex form. By mastering completing the square, students and professionals alike can gain a deeper appreciation for the elegance and power of quadratic equations. In the given example, rewriting $x^2 - 16x + 64 = -8$ as $(x - 8)^2 = -8$ showcases the direct application of this technique when dealing with perfect square trinomials, offering a clear pathway to understanding and solving such equations.

The process of completing the square, while sometimes appearing complex, becomes more intuitive with practice. The key is to remember the goal: to transform the quadratic expression into a perfect square trinomial. Once this is achieved, the equation can be easily solved or analyzed, revealing the underlying mathematical principles and relationships.

In conclusion, mastering completing the square not only equips you with a powerful tool for solving quadratic equations but also enhances your overall mathematical understanding and problem-solving skills. It is a cornerstone technique that bridges various mathematical concepts and opens doors to more advanced topics in algebra and beyond.