Solving $x^2 - 16x + 60 = -12$ By Completing The Square

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Hey guys! Today, we're going to tackle a classic quadratic equation using a method called "completing the square." It might sound intimidating, but trust me, it's a super useful technique for solving these types of problems. We'll break it down step-by-step so you can master it too. So, let's dive right into solving the equation x2−16x+60=−12x^2 - 16x + 60 = -12 by completing the square. This method is super handy for rewriting quadratic equations into a form that makes it easy to find the solutions, or roots. You know, those values of x that make the equation true. Completing the square is a powerful technique not just for solving equations, but also for understanding the structure of quadratic expressions and for graphing parabolas. So, stick with me, and let's get started!

Understanding Completing the Square

Before we jump into the nitty-gritty of this specific equation, let’s quickly recap what completing the square actually means. At its heart, completing the square is a method that transforms a quadratic expression (like x2+bx+cx^2 + bx + c) into a perfect square trinomial, plus or minus a constant. A perfect square trinomial is something you can factor into the form (x+a)2(x + a)^2 or (x−a)2(x - a)^2. Think of it like turning a messy puzzle into a neat, recognizable shape. The goal is to manipulate the equation so that we have a squared term on one side, which makes it much easier to solve for x. The basic idea behind completing the square is to manipulate a quadratic expression into the form (x+a)2+k(x + a)^2 + k or (x−a)2+k(x - a)^2 + k, where a and k are constants. This form is incredibly useful because it reveals the vertex of the parabola represented by the quadratic equation, and it makes solving for x straightforward. Essentially, we're trying to create a perfect square trinomial—an expression that can be factored into the form (x+a)2(x + a)^2 or (x−a)2(x - a)^2. This involves adding and subtracting a specific value to the original expression, which might seem a bit like magic at first, but it's rooted in solid algebraic principles.

Step-by-Step Solution

Okay, let's get our hands dirty and walk through the steps to solve x2−16x+60=−12x^2 - 16x + 60 = -12 using the completing the square method. We'll take it one step at a time, so don't worry if it seems a little confusing at first. We'll get there together! Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become with the process. So, grab your pencils, and let's get started!

Step 1: Move the Constant Term

First things first, we want to isolate the terms with x on one side of the equation. To do this, we'll move the constant term (the number without any x attached) to the right side. In our equation, that constant term is 60. So, let's subtract 60 from both sides of the equation:

x2−16x+60−60=−12−60x^2 - 16x + 60 - 60 = -12 - 60

This simplifies to:

x2−16x=−72x^2 - 16x = -72

Now we have the x2x^2 and x terms nicely isolated on the left side. This is a crucial first step because it sets us up for the next part of the process: creating that perfect square trinomial we talked about earlier. We're one step closer to unraveling this equation!

Step 2: Complete the Square

Now comes the heart of the method: completing the square. This is where we'll transform the left side of the equation into a perfect square trinomial. The trick is to add a specific number to both sides of the equation. But how do we find that magic number? Here's the formula: take half of the coefficient of the x term (the number in front of the x), square it, and that's the number we need to add. In our equation, the coefficient of the x term is -16. So, let's do the math:

  1. Take half of -16: -16 / 2 = -8
  2. Square the result: (-8)^2 = 64

So, 64 is the number we're going to add to both sides of the equation. This ensures that we maintain the balance of the equation while creating our perfect square. Let's add 64 to both sides:

x2−16x+64=−72+64x^2 - 16x + 64 = -72 + 64

Step 3: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial! That means it can be factored into the form (x−a)2(x - a)^2. In our case, the trinomial x2−16x+64x^2 - 16x + 64 factors into (x−8)2(x - 8)^2. Remember that -8 we calculated earlier when we took half of the coefficient of the x term? That's the number that goes inside the parentheses with x. On the right side of the equation, we can simplify -72 + 64 to -8. So, our equation now looks like this:

(x−8)2=−8(x - 8)^2 = -8

See how much simpler the equation looks now? We've successfully transformed it into a form where we have a squared term isolated on one side. This is the key to unlocking the solution!

Step 4: Take the Square Root

Now we're getting to the exciting part where we actually solve for x. To get rid of the square on the left side, we'll take the square root of both sides of the equation. But remember, when we take the square root, we need to consider both the positive and negative roots:

(x−8)2=±−8\sqrt{(x - 8)^2} = \pm\sqrt{-8}

This simplifies to:

x−8=±−8x - 8 = \pm\sqrt{-8}

Notice that we have the square root of a negative number, -8. This means our solutions will involve imaginary numbers. Don't worry, that's perfectly fine! We'll deal with that in the next step.

Step 5: Simplify the Square Root and Solve for x

Let's simplify that square root of -8. We can rewrite −8\sqrt{-8} as 8∗−1\sqrt{8} * \sqrt{-1}. Remember that −1\sqrt{-1} is defined as the imaginary unit, i. And 8\sqrt{8} can be simplified to 222\sqrt{2}. So, we have:

x−8=±22ix - 8 = \pm 2\sqrt{2}i

Now, to isolate x, we'll add 8 to both sides of the equation:

x=8±22ix = 8 \pm 2\sqrt{2}i

And there we have it! Our solutions are complex numbers, which means they have both a real part (8) and an imaginary part (22i2\sqrt{2}i and −22i-2\sqrt{2}i).

Final Answer

So, the solutions to the equation x2−16x+60=−12x^2 - 16x + 60 = -12 are:

x=8+22ix = 8 + 2\sqrt{2}i and x=8−22ix = 8 - 2\sqrt{2}i

These are complex solutions, which means they involve the imaginary unit i. This is a perfectly valid outcome, and it tells us that the parabola represented by the quadratic equation does not intersect the x-axis in the real number plane. That's a wrap, guys! We've successfully solved the equation by completing the square. I hope you found this explanation helpful. Remember, the key to mastering this technique is practice, so don't hesitate to try more examples. Keep up the great work!

Key Takeaways

Completing the square is a fundamental technique in algebra, and mastering it can significantly enhance your problem-solving skills. Beyond solving quadratic equations, it's a powerful tool for understanding the structure and properties of parabolas, as well as for various other mathematical applications. Let's recap some of the key takeaways from this process.

  • Isolate the x Terms: The first crucial step is to isolate the x² and x terms on one side of the equation. This sets the stage for completing the square. Make sure to move any constant terms to the other side.
  • Find the Magic Number: The number you need to add to both sides to complete the square is found by taking half of the coefficient of the x term and then squaring it. This is the core of the method, so make sure you understand this step thoroughly.
  • Factor the Perfect Square: Once you've added the magic number, the quadratic expression on one side should factor into a perfect square trinomial, which can be written in the form (x + a)² or (x - a)². This is a crucial simplification that makes the equation much easier to solve.
  • Consider Both Roots: When taking the square root of both sides, remember to consider both the positive and negative roots. This is essential for finding all possible solutions to the equation.
  • Deal with Complex Solutions: If you encounter the square root of a negative number, don't be alarmed! This simply means the solutions are complex numbers, which involve the imaginary unit i. Complex solutions are perfectly valid and often arise in various mathematical contexts.
  • Practice Makes Perfect: Like any mathematical technique, mastering completing the square requires practice. Work through various examples, and don't hesitate to revisit the steps and concepts as needed.

Why Completing the Square Matters

You might be wondering, with so many methods for solving quadratic equations (like factoring and the quadratic formula), why bother learning completing the square? Well, there are several compelling reasons why this technique is worth mastering. It's more than just a way to find solutions; it's a fundamental tool that provides deeper insights into quadratic equations and their graphical representations.

  • Understanding the Vertex Form: Completing the square allows you to rewrite a quadratic equation in vertex form: y=a(x−h)2+ky = a(x - h)^2 + k. This form directly reveals the vertex of the parabola, which is the point (h, k). The vertex is a crucial feature of a parabola, as it represents the maximum or minimum point of the curve. Knowing the vertex form makes graphing the parabola much easier.
  • Deriving the Quadratic Formula: Did you know that the quadratic formula itself can be derived by completing the square on the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0? This demonstrates the power and versatility of the completing the square method. It's not just a standalone technique; it's a foundational concept that underpins other important results in algebra.
  • Solving Optimization Problems: Completing the square can be used to solve optimization problems, where you need to find the maximum or minimum value of a quadratic function. By rewriting the function in vertex form, you can easily identify the vertex, which corresponds to the optimal value. This has applications in various fields, such as physics, engineering, and economics.
  • Building a Strong Foundation: Mastering completing the square strengthens your algebraic skills and problem-solving abilities. It reinforces concepts like factoring, manipulating equations, and working with square roots. This solid foundation will benefit you in more advanced math courses and in various real-world applications.

In conclusion, completing the square is a valuable tool in your mathematical arsenal. It not only provides a way to solve quadratic equations but also enhances your understanding of quadratic functions and their properties. So, keep practicing and exploring its applications, and you'll find it to be a powerful and versatile technique.