Problem 69 - Algebra 2

Question

Solve each equation, and locate the complex solutions in the complex plane. $$ \frac{4}{5} x^{2}+1=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \pm \frac{\sqrt{5}}{2}i\) located on the imaginary axis.

Step 1: Isolate the quadratic term

First, rewrite the original equation to isolate the term involving \(x^2\). The given equation is \(\frac{4}{5}x^2 + 1 = 0\). Subtract 1 from both sides to get \(\frac{4}{5}x^2 = -1\).

Step 2: Clear the fraction

Multiply both sides by \(\frac{5}{4}\) to eliminate the fraction: \(x^2 = -\frac{5}{4}\).

Step 3: Solve for \(x\) using the square root

Take the square root of both sides of the equation \(x^2 = -\frac{5}{4}\). The solutions involve the imaginary unit \(i\). Thus, \(x = \pm \sqrt{-\frac{5}{4}} = \pm i \sqrt{\frac{5}{4}}\).

Step 4: Simplify the expression

Simplify \(\pm i \sqrt{\frac{5}{4}}\) to \(\pm \frac{\sqrt{5}}{2}i\). The solutions are \(x = \frac{\sqrt{5}}{2}i\) and \(x = -\frac{\sqrt{5}}{2}i\).

Step 5: Locate the solutions in the complex plane

In the complex plane, the solution \(x = \frac{\sqrt{5}}{2}i\) is located on the positive imaginary axis at a distance \(\frac{\sqrt{5}}{2}\) from the origin. The solution \(x = -\frac{\sqrt{5}}{2}i\) is located on the negative imaginary axis at the same distance.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions set to zero and are in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations are known for having up to two solutions, often referred to as roots. These solutions can be real or complex numbers, depending on the discriminant (\(b^2 - 4ac\)). When the discriminant is negative, the roots become complex numbers.

• In real-world terms, the quadratic equation describes a parabola in two-dimensional space.
• If the parabola crosses the x-axis, the solutions are real.
• If it doesn't cross the x-axis, the solutions are complex.

Whether real or complex, solving a quadratic equation can be approached using various methods including factoring, completing the square, or applying the quadratic formula. These techniques are foundational for algebra and are crucial in various branches of mathematics.

Complex Plane

The complex plane, also referred to as the Argand plane, is a way to visualize complex numbers. Instead of using two separate numbers to describe an ordered pair, a complex number combines two numbers: a real part and an imaginary part.

• The horizontal axis represents the real part.
• The vertical axis represents the imaginary part.

The complex solutions to our quadratic equation example lie purely on this imaginary axis since they have no real component. Each complex number is a unique point in this plane, providing a powerful tool to visualize and operate on complex numbers. By plotting these numbers, mathematicians can better understand behavior and relationships in complex equations.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the real number line into the complex plane. It is defined by the property \(i^2 = -1\). This definition is crucial for handling the square roots of negative numbers, which are not possible within the set of real numbers alone.

• Using \(i\), one can express any negative square root in terms of a real number multiplied by \(i\).
• In the equation \(x^2 = -\frac{5}{4}\), the solutions involve \(i\) because the term \(-\frac{5}{4}\) is negative.

Understanding and using the imaginary unit can open doors to solving otherwise impossible mathematical quandaries. This makes it an indispensable tool in both theoretical mathematics and its applications.

Square Root Method

The square root method is a straightforward and effective technique for solving quadratic equations that lack a linear \(bx\) term. Using it requires isolating the quadratic variable, taking the square root of both sides, and parsing solutions.
Steps in our context:

• First, isolate \(x^2\) to one side of the equation: this step involves basic algebra to get \(x^2 = -\frac{5}{4}\).
• Next, tackle the square root of both sides. Since the right side is negative, introduce \(i\), resulting in \(x = \pm i \sqrt{\frac{5}{4}}\).
• Finally, simplify the expression, giving solutions \(x = \frac{\sqrt{5}}{2}i\) and \(x = -\frac{\sqrt{5}}{2}i\).

This method beautifully illustrates the transition from real to complex solutions, showcasing the significance of imaginary numbers in finding roots of equations that cannot be resolved on the real number line alone.