Problem 69 - Elementary Algebra

Question

\(3 x^{2}=-9\)

Step-by-Step Solution

Answer

x = \pm \sqrt{3}i

Step 1 - Divide both sides by 3

To isolate the squared term, divide both sides of the equation by 3: \[ \frac{3x^2}{3} = \frac{-9}{3} \] This simplifies to: \[ x^2 = -3 \]

Step 2 - Take the square root of both sides

To solve for x, take the square root of both sides of the equation. Remember that taking the square root of a negative number involves imaginary numbers: \[ x = \pm \sqrt{-3} \] This can be simplified to: \[ x = \pm \sqrt{3}i \]

Key Concepts

Imaginary Numbers

When you try to take the square root of a negative number, you encounter an imaginary number. Imaginary numbers are numbers that give a negative result when squared.
To handle such cases, mathematicians introduced the imaginary unit denoted as i.
This unit is defined as \(\text{\text i} \) where \(\text{\text i}^{2} = -1\).
For example, the square root of -3 can be written as \(\text{\text i}\) times the square root of 3, or \(\text{\sqrt{-3} = \text \sqrt{3} \text{\ i}}\). This is how we express solutions involving imaginary numbers.
Imaginary numbers are particularly useful in various fields, especially in engineering and physics.

Square Root

The square root function essentially reverses the squaring operation. Taking the square root of a number means finding a value that, when multiplied by itself, gives the original number.
For instance: \(\sqrt{9} = 3 \) because \((3 \times 3 = 9) \).
It's important to remember that every positive number has two square roots: a positive and a negative one. That's why when you take the square root of both sides of an equation, you write \( x = \pm \sqrt{(\text{number})} \). The \(\pm\) symbol stands for both possibilities.
When dealing with a negative number under the square root, you must incorporate imaginary numbers. For our example: \[ x = \pm \sqrt{-3} \] simplifies to \[ x = \pm \sqrt{3} \text{i} \].

Isolating Variables

Isolating the variable means manipulating the equation to get the variable by itself on one side of the equation. This is often the first step in solving for the variable.
In our exercise: \[ 3x^2 = -9 \], you want to isolate \(x^{2}\). Divide both sides by 3: \[ \frac{3x^2}{3} = \frac{-9}{3} \]
This simplifies to: \[ x^2 = -3 \]
Always aim to rearrange the equation step by step, following basic arithmetic and algebraic rules, until the variable you need to solve for is isolated. This makes the equation simpler to solve.