Problem 13 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \frac{-6 \pm \sqrt{-72}}{3} $$

Step-by-Step Solution

Answer

The simplified form is \(-2 \pm 2i\sqrt{2}\).

Step 1: Identify the Imaginary Unit

Recall that the imaginary unit is defined as \( i = \sqrt{-1} \). This means that \( \sqrt{-72} = \sqrt{-1 \times 72} = \sqrt{-1} \times \sqrt{72} = i \sqrt{72} \).

Step 2: Simplify the Square Root

Simplify \( \sqrt{72} \). Note that \( 72 = 36 \times 2 \), so \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \). Therefore, the expression becomes \( \sqrt{-72} = 6i\sqrt{2} \).

Step 3: Substitute and Simplify the Expression

Substitute \( 6i\sqrt{2} \) for \( \sqrt{-72} \) into the original expression. This gives us \( \frac{-6 \pm 6i\sqrt{2}}{3} \).

Step 4: Simplify Each Part Separately

Separate the expression into two parts: \( \frac{-6}{3} \pm \frac{6i\sqrt{2}}{3} \). Simplifying each fraction, we get \( -2 \pm 2i\sqrt{2} \).

Step 5: Final Simplified Form

The expression \( \frac{-6 \pm \sqrt{-72}}{3} \) simplifies to \( -2 \pm 2i\sqrt{2} \), which represents two possible results: \( -2 + 2i\sqrt{2} \) and \( -2 - 2i\sqrt{2} \).

Key Concepts

Complex Numbers

A complex number is like a mysterious friend in mathematics, combining familiar real numbers with an intriguing twist, imaginary numbers. It is often written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\).
This concept becomes particularly exciting when we deal with square roots of negative numbers, as they don't have solutions in the realm of real numbers. By using complex numbers, we can solve these previously unsolvable problems.

• The introduction of complex numbers allows for a broader understanding in mathematics, offering solutions in electrical engineering, physics, and more.
• They provide a way to handle situations that seem impossible with only real numbers.

The key takeaway is that with complex numbers, we open the doors to a whole new dimension of problem-solving.

Square Roots

Square roots are fundamental in mathematics, as they represent a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25.
But what about square roots of negative numbers? This is where the imaginary unit \(i\) comes in handy. The square root of a negative number is expressed using \(i\).

• For example, \(\sqrt{-72}\) can be rewritten as \(\sqrt{-1 \times 72}\), which simplifies to \(i\sqrt{72}\).
• Once understood, this transformation of square roots involving negative numbers becomes quite manageable.

This understanding allows us to handle more complex expressions using both real and imaginary parts.

Algebraic Expressions

Algebraic expressions are like little puzzles where numbers and variables come together under operations like addition, subtraction, multiplication, and division. When imaginary units enter these expressions, they add another layer of complexity.
In our exercise, we saw how the expression \(\frac{-6 \pm \sqrt{-72}}{3}\) was simplified using the imaginary unit. We dealt with both the real and imaginary parts separately, making the process manageable.

• This separation allows us to individually simplify terms and eventually combine them into a complete solution.
• For example, \(\frac{-6}{3}\) and \(\frac{6i\sqrt{2}}{3}\) simplify separately before the two parts are combined.

Handling algebraic expressions involving complex numbers involves simplifying each part with care, opening new possibilities for solving equations that initially seemed unsolvable.