Problem 68 - Algebra 2

Question

Solve each equation, and locate the complex solutions in the complex plane. $$ \frac{2}{3} x^{2}+30=0 $$

Step-by-Step Solution

Answer

The complex solutions are \( x = 3\sqrt{5}i \) and \( x = -3\sqrt{5}i \) on the imaginary axis.

Step 1: Isolate the quadratic term

First, isolate the quadratic term by subtracting 30 from both sides of the equation: \( \frac{2}{3}x^2 = -30 \).

Step 2: Eliminate the fraction

To clear the fraction, multiply every term in the equation by 3, which gives: \( 2x^2 = -90 \).

Step 3: Solve for \( x^2 \)

Next, divide both sides by 2 to solve for \( x^2 \): \( x^2 = -45 \).

Step 4: Solve for \( x \) using complex numbers

To find \( x \), take the square root of both sides. Remember, \( \sqrt{-1} = i \), the imaginary unit. Therefore, \( x = \pm \sqrt{-45} \), which simplifies to \( x = \pm \sqrt{45i^2} \). Further simplification leads us to \( x = \pm 3\sqrt{5}i \).

Step 5: Locate in the complex plane

The solutions \( x = 3\sqrt{5}i \) and \( x = -3\sqrt{5}i \) locate on the vertical imaginary axis. The points are \( (0, 3\sqrt{5}) \) and \( (0, -3\sqrt{5}) \) on the complex plane.

Key Concepts

Quadratic Equations

Quadratic equations are a core part of algebra, characterized by expressions set to zero in the form of \( ax^2 + bx + c = 0 \). The equation features a variable squared and usually involves real and complex numbers as potential solutions. These equations appear frequently in a variety of mathematical contexts and often require specific strategies to solve.

Here's a gentle breakdown:

• The highest power of the variable, \( x \), is 2, which makes it quadratic.
• They can feature integer, fractional, or even irrational coefficients.
• Solutions can be real, or if the discriminant is negative, complex.

This particular exercise involves a quadratic term where all other variables are constants, allowing us to determine solutions using standard algebraic manipulation. They often lead to deeper explorations into the nature of numbers when extended to complex solutions.

Complex Plane

The complex plane is an essential concept in understanding complex numbers. It's a two-dimensional plane where each point represents a complex number.

Here's what you need to know:

• The horizontal axis, known as the real axis, corresponds to the real part of the complex number.
• The vertical axis, known as the imaginary axis, represents the imaginary part of the number.

For example, in this problem, the solutions \( x = 3\sqrt{5}i \) and \( x = -3\sqrt{5}i \) are located on the imaginary axis, precisely because their real parts are zero. Thus, they are placed at points along the vertical, with their values found by moving \( 3\sqrt{5} \) units up or down from the origin.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental building block in the realm of complex numbers. Defined by the property \( i^2 = -1 \), it allows us to extend the real number system into the complex one.

Essential components include:

• \( i \) is used to represent the square root of negative one.
• It creates a bridge to solving equations where negative numbers appear under the square root.

In context, when deriving solutions such as \( x^2 = -45 \), we use \( i \) to assist in expressing the result, leading to \( x = \pm 3\sqrt{5}i \). Without the imaginary unit, finding square roots of negatives would remain impossible, hampering further algebraic applications.

Solutions of Equations

When solving equations, the main objective is to find the values that satisfy the given expression or condition. In the context of quadratic equations that yield complex solutions, the approach involves acknowledging and working with the imaginary component.

Here's the solution process overview:

• Isolate and rearrange as needed to handle terms systematically.
• Use algebraic operations to simplify until it's possible to apply the square root.
• Recognize the use of \( i \) in obtaining and expressing roots of negative numbers.

In our exercise, solutions are complex, \( x = 3\sqrt{5}i \) and \( x = -3\sqrt{5}i \). These values confirm the equation is satisfied upon substitution and display the elegance of utilizing complex forms.