Problem 73 - Beginning and Intermediate Algebra

Question

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-100 $$

Step-by-Step Solution

Answer

The solutions are \( x = 10i \) and \( x = -10i \).

Step 1: Move constant to the right side of the equation

The equation is already in the form: \[ x^2 = -100 \]No need to move any terms as it is already prepared for solving by taking the square root.

Step 2: Take the square root of both sides

To isolate \( x \), take the square root of both sides of the equation: \[ x = \pm \sqrt{-100} \]Remember that taking the square root introduces both positive and negative roots.

Step 3: Simplify the square root of a negative number

Recognize that \( \sqrt{-100} \) can be written in terms of the imaginary unit \( i \), where \( \sqrt{-1} = i \) and \( \sqrt{100} = 10 \): \[ \sqrt{-100} = \sqrt{100 \cdot (-1)} = \sqrt{100} \cdot \sqrt{-1} = 10i \]Thus, the solutions are: \[ x = \pm 10i \]

Key Concepts

solving quadratic equations

Quadratic equations are equations in the form of \(ax^2 + bx + c = 0\). They can have real or complex solutions.
To solve, we generally follow these steps:

• Move all terms to one side, setting the equation to zero.
• Factorize, complete the square, or use the quadratic formula to find solutions.

When solving, remember that the equation \(x^2 = n\) can be solved by taking the square root of both sides. If \(n\) is negative, this introduces imaginary numbers.

imaginary numbers

Imaginary numbers are numbers that when squared give a negative result. The basic imaginary unit is represented as \(i\), where \(i = \sqrt{-1}\).
For instance:

• The square of \(i\) is \(i^2 = -1\).
• Any real number multiplied by \(i\) is an imaginary number.

This concept is crucial when we encounter square roots of negative numbers in quadratic equations.

square roots of negative numbers

The square root of a negative number involves imaginary numbers. For example, the square root of -100 can be simplified as follows:
\[ \sqrt{-100} = \sqrt{100 \cdot (-1)} = \sqrt{100} \cdot \sqrt{-1} = 10i \]
Thus, \( \sqrt{-100} \) is 10i. When solving equations where \(x^2\) is equal to a negative number, the solutions will involve \(i\). In our exercise, that gave us \(\pm 10i\).
This ensures that we understand the introduction of \(i\) when dealing with such problems.