Problem 101 - Algebra for College Students

Question

Some of the solution sets for quadratic equations in the next sections will contain complex numbers such as \((-4+\sqrt{-12}) / 2\) and \((-4-\sqrt{-12}) / 2\). We can simplify the first number as follows. $$ \begin{array}{r} \frac{-4+\sqrt{-12}}{2}=\frac{-4+i \sqrt{12}}{2}= \\ \frac{-4+2 i \sqrt{3}}{2}=\frac{2(-2+i \sqrt{3})}{2}=-2+i \sqrt{3} \end{array} $$ Simplify each of the following complex numbers. (a) \(\frac{-4-\sqrt{-12}}{2}\) (b) \(\frac{6+\sqrt{-24}}{4}\) (c) \(\frac{-1-\sqrt{-18}}{2}\) (d) \(\frac{-6+\sqrt{-27}}{3}\) (e) \(\frac{10+\sqrt{-45}}{4}\) (f) \(\frac{4-\sqrt{-48}}{2}\)

Step-by-Step Solution

Answer

(a) \(-2 - i\sqrt{3}\), (b) \(\frac{3}{2} + \frac{i\sqrt{6}}{2}\), (c) \(-\frac{1}{2} - \frac{3i\sqrt{2}}{2}\), (d) \(-2 + i\sqrt{3}\), (e) \(\frac{5}{2} + \frac{3i\sqrt{5}}{4}\), (f) \(2 - 2i\sqrt{3}\).

Step 1: Simplify Part (a)

Start with the expression \( \frac{-4 - \sqrt{-12}}{2} \). Recognize \( \sqrt{-12} = i\sqrt{12} = i\cdot 2\sqrt{3} \). Substitute \( i\cdot 2\sqrt{3} \) for \( \sqrt{-12} \) to get \( \frac{-4 - 2i\sqrt{3}}{2} \). Simplify it by dividing both terms in the numerator by 2, resulting in \( -2 - i\sqrt{3} \).

Step 2: Simplify Part (b)

Consider the expression \( \frac{6 + \sqrt{-24}}{4} \). Realize that \( \sqrt{-24} = i\sqrt{24} = i\cdot 2\sqrt{6} \). Plug \( i\cdot 2\sqrt{6} \) for \( \sqrt{-24} \) to rewrite the expression as \( \frac{6 + 2i\sqrt{6}}{4} \). Divide each term by 4 to get \( \frac{3}{2} + \frac{i\sqrt{6}}{2} \).

Step 3: Simplify Part (c)

Start with \( \frac{-1 - \sqrt{-18}}{2} \). Notice that \( \sqrt{-18} = i\sqrt{18} = i\cdot 3\sqrt{2} \). Substitute \( i\cdot 3\sqrt{2} \) for \( \sqrt{-18} \) to obtain \( \frac{-1 - 3i\sqrt{2}}{2} \). This results in \( -\frac{1}{2} - \frac{3i\sqrt{2}}{2} \) after simplification.

Step 4: Simplify Part (d)

Take the fraction \( \frac{-6 + \sqrt{-27}}{3} \). Recognize \( \sqrt{-27} = i\sqrt{27} = i\cdot 3\sqrt{3} \). Substituting \( i\cdot 3\sqrt{3} \) yields \( \frac{-6 + 3i\sqrt{3}}{3} \). Simplify by dividing through by 3, resulting in \( -2 + i\sqrt{3} \).

Step 5: Simplify Part (e)

Begin with \( \frac{10 + \sqrt{-45}}{4} \). Here, \( \sqrt{-45} = i\sqrt{45} = i\cdot 3\sqrt{5} \). Plug this into the expression to form \( \frac{10 + 3i\sqrt{5}}{4} \). After simplification, it becomes \( \frac{5}{2} + \frac{3i\sqrt{5}}{4} \).

Step 6: Simplify Part (f)

Consider the expression \( \frac{4 - \sqrt{-48}}{2} \). Here \( \sqrt{-48} = i\sqrt{48} = i\cdot 4\sqrt{3} \). Substitute to get \( \frac{4 - 4i\sqrt{3}}{2} \), which simplifies to \( 2 - 2i\sqrt{3} \).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. They play a crucial role in algebra and often appear in various mathematical problems. The solutions to quadratic equations can be found using different methods:

\( \bullet \) **Factoring**: Breaking down the equation into simpler terms that can be solved with basic algebra.
\( \bullet \) **Completing the square**: A method involving transforming the equation into a perfect square trinomial.
\( \bullet \) **Quadratic formula**: An explicit formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) that provides the solutions for any quadratic equation.

Quadratic equations often yield two possible solutions, represented by \( \pm \sqrt{b^2 - 4ac} \), also known as the discriminant. If the discriminant is negative, the solutions involve complex numbers.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental concept in complex numbers. It is defined by the property \( i^2 = -1 \). Imaginary numbers extend the real number system to solve problems involving negative square roots.

Key points to understand about the imaginary unit include:

\( \bullet \) **Basic operations**: Imaginary numbers can be added, subtracted, multiplied, and divided like real numbers, with the crucial difference being \( i^2 = -1 \). This property is used to simplify expressions like \( \sqrt{-12} = i \sqrt{12} \).
\( \bullet \) **Complex numbers**: Combining real numbers and imaginary numbers gives us complex numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is the real part and \( bi \) is the imaginary part.

The imaginary unit enables solving equations and expressing solutions that involve the square root of negative numbers.

Simplification of Expressions

Simplifying expressions involving complex numbers is essential for ease of understanding and further calculations. The process typically includes:

\( \bullet \) **Identifying the imaginary component**: Recognize when a square root involves a negative number, such as \( \sqrt{-12} = i \cdot \sqrt{12} \).
\( \bullet \) **Factorization**: Breaking down expressions under the square root to their simplest form. For instance, in \( \sqrt{12} \), recognizing \( 12 = 4 \times 3 \) helps simplify it to \( 2\sqrt{3} \).
\( \bullet \) **Algebraic manipulation**: Such as dividing both terms of a fraction to simplify expressions, ensuring the solution is in its simplest form, like reducing \( \frac{-4 - 2i\sqrt{3}}{2} \) to \( -2 - i\sqrt{3} \).

Simplification allows expressions to be dealt with more easily, often making them more manageable in calculations and interpretations.