Solving (x-4)^2 = -49: Exact & Approximate Solutions

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Hey guys! Let's dive into solving the equation (x-4)^2 = -49. This is a fun one because it involves dealing with a negative number under a square, which means we'll be venturing into the realm of imaginary numbers. Don't worry, it's not as scary as it sounds! We'll break it down step by step to find both the exact and approximate solutions, rounding to three decimal places where necessary. So, buckle up and let's get started!

Understanding the Problem

First things first, let's understand what the equation (x-4)^2 = -49 is telling us. We have a squared term, (x-4)^2, which equals -49. Now, remember that squaring any real number (positive or negative) always results in a non-negative number. Think about it: 2^2 = 4 and (-2)^2 = 4. So, how can a square be negative? That's where imaginary numbers come into play!

Key Concept: Imaginary Numbers

The imaginary unit, denoted by 'i', is defined as the square root of -1 (i = √-1). This is crucial because it allows us to work with square roots of negative numbers. For example, √-9 can be written as √(9 * -1) = √9 * √-1 = 3i. Got it? Awesome!

Why This Problem Matters

You might be wondering, β€œWhy do I even need to know this?” Well, understanding imaginary numbers and how to solve equations like this one is super important in various fields, including:

  • Electrical Engineering: Analyzing AC circuits often involves complex numbers (a combination of real and imaginary numbers).
  • Quantum Mechanics: Describing the behavior of particles at the atomic level relies heavily on complex numbers.
  • Signal Processing: Imaginary numbers are used to represent and manipulate signals in various applications, like audio and image processing.
  • Mathematics: This is a fundamental concept in algebra and calculus, which forms the basis for many other mathematical fields.

So, grasping this concept opens doors to understanding more advanced topics and real-world applications. Let’s get back to solving our equation!

Finding the Exact Solutions

Alright, let's tackle the equation (x-4)^2 = -49 and find those exact solutions. Here’s how we'll do it:

  1. Take the Square Root of Both Sides:

    To get rid of the square on the left side, we take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots. This gives us:

    √((x-4)^2) = ±√(-49)

x - 4 = ±√(-49)

  1. Introduce the Imaginary Unit:

    Now, we have √(-49), which we can rewrite using the imaginary unit 'i'. Since √(-49) = √(49 * -1) = √49 * √-1, we get:

x - 4 = Β±7i

  1. Isolate x:

    To solve for x, we simply add 4 to both sides of the equation:

x = 4 Β± 7i

Boom! We've found our exact solutions. They are x = 4 + 7i and x = 4 - 7i. These are complex numbers, with a real part (4) and an imaginary part (7i and -7i).

A Quick Recap on Exact Solutions

  • We started with (x-4)^2 = -49.
  • Took the square root of both sides: x - 4 = ±√(-49).
  • Introduced the imaginary unit: x - 4 = Β±7i.
  • Isolated x: x = 4 Β± 7i.

So, the exact solutions are:

  • x = 4 + 7i
  • x = 4 - 7i

Now, let's move on to finding the approximate solutions. Since our solutions are complex numbers, the real part is already expressed exactly (4). However, if we needed to approximate a square root or other irrational number, we would round to three decimal places as required. In this case, there is nothing more to approximate as the solution is already exact.

Finding Approximate Solutions

In this particular case, the solutions we found (x = 4 + 7i and x = 4 - 7i) are exact. The real part, 4, is already a precise value, and the imaginary part, 7i, is also exact since 7 is an integer and 'i' represents the square root of -1. When a problem asks for approximate solutions to three decimal places, it usually implies dealing with irrational numbers that need rounding, such as square roots of non-perfect squares (like √2) or transcendental numbers like Ο€.

However, since our solutions are complex numbers with integer coefficients, there's no need for further approximation. We can confidently state that the approximate solutions are the same as the exact solutions:

  • x β‰ˆ 4 + 7i
  • x β‰ˆ 4 - 7i

Why No Further Approximation is Needed

To clarify, let's think about scenarios where approximation is crucial:

  • Irrational Numbers: Numbers like √2 (approximately 1.414) or Ο€ (approximately 3.142) cannot be expressed as a simple fraction and have infinite non-repeating decimal expansions. We often round these for practical purposes.
  • Complex Calculations: If we had a solution like x = (√5 + 3i) / 2, we'd need to approximate √5 to a certain number of decimal places to get an approximate value for x.

In our case, the imaginary unit 'i' is exact by definition (√-1), and the other numbers are integers. Therefore, our solutions are already in their simplest, most precise form.

Visualizing the Solutions

Okay, so we’ve found our solutions, but what do they actually mean? Since we're dealing with complex numbers, we can't just plot them on a regular number line. Instead, we use something called the complex plane.

The Complex Plane

The complex plane is similar to the Cartesian plane (the regular x-y plane) but with a twist. The horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. So, a complex number a + bi is plotted as the point (a, b).

For our solutions, x = 4 + 7i and x = 4 - 7i:

  • 4 + 7i would be plotted at the point (4, 7).
  • 4 - 7i would be plotted at the point (4, -7).

Visual Representation:

Imagine a graph where the x-axis is labeled