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Chapter 8: Problem 25

Find all complex-number solutions. $$ (x+1)^{2}=-9 $$

Short Answer

Expert verified
The solutions are \(x = 3i - 1\) and \(x = -3i - 1\).

Step by step solution

01

- Isolate the squared term

Start by isolating the squared term on one side of the equation. In this case, \[(x+1)^2 = -9\].
02

- Apply the square root property

Taking the square root of both sides will give us \[\sqrt{(x+1)^2} = \sqrt{-9}\]. Remember, the square root of a negative number involves imaginary numbers. So, we have \[|x + 1| = \pm 3i\], where \(i\) is the imaginary unit.
03

- Solve for x

Solve for \(x\) by breaking it into two separate equations: \[x + 1 = 3i\] and \[x + 1 = -3i\].Solving each, we get: \[x = 3i - 1\] and \[x = -3i - 1\].

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Key Concepts

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

Isolating Squared Terms
When solving complex-number equations, the first step often involves isolating the squared term. This means you want to get the squared term by itself on one side of the equation.
In our example, the starting equation is \((x+1)^{2} = -9\).
To isolate the term \((x+1)^{2}\), it's already nicely placed on one side of the equation, so no further isolating is necessary for this particular problem.
In general, you might sometimes need to move other terms from one side to the other to achieve this isolation.
  • **Tip:** Always perform operations that keep the equation balanced. What you do to one side, do to the other.
Understanding this will help simplify the rest of the steps, making it easier to solve for the variable.
Square Root Property
Next, we use the square root property, which helps us eliminate the squared term. Taking the square root of both sides of the equation is a common method:
From our isolated term, we take the square root of both sides: \((x+1)^{2} = -9\).
This results in \(\text{sqrt}((x+1)^{2}) = \text{sqrt}(-9)\).
Since we know that \(\text{sqrt}((x+1)^{2})\) is just \(|x + 1|\), we get \(|x + 1| = \text{sqrt}(-9)\).
Importantly, remember the square root of a negative number involves imaginary numbers. Hence, we write this as \(|x + 1| = \text{plus or minus} 3i\), where \(i\) is the imaginary unit representing \(\sqrt{-1}\).
  • **Insight:** Whenever you see the term \(\sqrt{-n}\), think of it as multiplying \(\sqrt{n}\) by \(i\).
Imaginary Numbers
Imaginary numbers make it possible to handle the square root of negative numbers. The fundamental imaginary unit is \(i\), where \(i = \sqrt{-1}\). These are the key points for understanding imaginary numbers:
  • **Imaginary Unit:** \(i\) itself is defined as \(\sqrt{-1}\).
  • **Complex Numbers:** Numbers like \(a + bi\), where \(a\) and \(b\) are real numbers and \(bi\) is the imaginary part, are called complex numbers.
  • **Operations:** When performing operations like addition or subtraction involving \(i\), treat \(i\) just like you would a variable, but remember \(i^2 = -1\).
In our example: From \( |x + 1| = \pm 3i\), we break it down into two separate equations:
1. \(x + 1 = 3i \rightarrow x = 3i - 1\)
2. \(x + 1 = -3i \rightarrow x = -3i - 1\)
Both solutions \(3i - 1\) and \(-3i - 1\) are complex numbers, presenting a combination of real and imaginary parts.

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