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Chapter 1: Problem 61

Solve each equation in the complex number system. $$ x^{2}-16=0 $$

Short Answer

Expert verified
x = \pm 4

Step by step solution

01

Identify the equation form

The given equation is a quadratic equation of the form: \[ x^2 - 16 = 0 \]
02

Move constant to the other side

Add 16 to both sides of the equation to isolate the quadratic term: \[ x^2 = 16 \]
03

Take the square root of both sides

To solve for x, take the square root of both sides of the equation. Remember that the square root of a number has both positive and negative roots: \[ x = \pm \sqrt{16} \]
04

Simplify the square root

Simplify the square root of 16: \[ \sqrt{16} = 4 \] So the equation becomes: \[ x = \pm 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equations
A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). The equation from the exercise, \(x^2 - 16 = 0\), fits this form because it can be rewritten as \(x^2 + 0x - 16 = 0\).

Quadratic equations often emerge in several contexts such as physics, engineering, and finance. Quadratic equations have different methods of solutions like:
  • Factoring
  • Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  • Completing the square
  • Graphical methods
In our exercise, we solve it through a straightforward method: isolating the quadratic term \(x^2\) and taking the square root.
solving equations
Solving equations means finding all possible values of the variable that make the equation true. In this case, we find values of \(x\) that satisfy \(x^2 - 16 = 0\). Here's how:

First, isolate the quadratic term by moving the constant to the other side. Here, we add 16 to both sides to get \(x^2 = 16\).

Next, solve for the variable by doing the opposite of squaring: taking the square root.

Remember, solving involves step-by-step simplification:
  • Identifying the form of the equation
  • Isolating the variable
  • Applying necessary mathematical operations like the square root
  • Simplifying the result
This step-by-step approach ensures that you systematically find all possible solutions.
square root
Taking the square root of a number means finding a value which, when squared, gives the original number. For any positive number \(k\), \(\sqrt{k}\) has two values: positive and negative. So \(\sqrt{16}\) returns both \(4\) and \(-4\).

In equations, this means if \(x^2 = k\), then \(x = \pm \sqrt{k}\), symbolizing two solutions. For instance, in our problem: \[ \text{If } x^2 = 16, \text{ then } x = \pm \sqrt{16} \text{ which means } x = 4 \text{ and } x = -4. \] This understanding is critical in the context of quadratic equations, ensuring all possible solutions, including negative and complex, are considered.

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