Problem 19 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-18$$

Step-by-Step Solution

Answer

No real solutions; solutions are \(x = \pm 3i\sqrt{2}\).

Step 1: Identify Equation Type

The given equation is \(x^2 = -18\). This is a quadratic equation because the highest power of \(x\) is 2.

Step 2: Analyze Possibility for Real Solutions

Quadratic equations of the form \(x^2 = a\) have real solutions only if \(a \geq 0\). Here, since \(a = -18\), which is less than zero, there are no real solutions.

Step 3: Determine Nature of Solutions

Since the number on the right side of the equation is negative, this equation has imaginary solutions. We can express the solutions using imaginary numbers.

Step 4: Find the Imaginary Solutions

To solve \(x^2 = -18\), take the square root of both sides. We get \(x = \pm \sqrt{-18}\). This simplifies to \(x = \pm i\sqrt{18}\), where \(i\) is the imaginary unit.

Step 5: Simplify the Solutions Further

Since \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\), the solutions become \(x = \pm 3i\sqrt{2}\).

Key Concepts

Imaginary Solutions

When dealing with quadratic equations like the one given in the exercise, if the result on one side of the equation is negative, the solutions turn out to be non-real or imaginary. An imaginary number involves the square root of a negative number. The imaginary unit, denoted as "\(i\)", is used to represent the square root of -1. This unit is fundamental when expressing imaginary solutions.Here’s how to find imaginary solutions step by step:

• Consider the equation \(x^2 = -18\).
• We know that taking the square root of a negative number involves "\(i\)".
• The solution becomes \(x = \pm \sqrt{-18}\).
• Since \(-18 = -1 \times 18\), it can be expressed as \(\pm i\sqrt{18}\).
• Simplifying further using properties of square roots gives \(x = \pm 3i\sqrt{2}\).

Imaginary solutions are crucial in problems where no real solutions can be found, allowing them to open new dimensions within mathematical reasoning.

Graphical Representation

Understanding imaginary solutions visually can be a bit challenging. When we graph a regular quadratic equation, such as \(x^2 = c\) where \(c\) is positive, it depicts a parabola that intersects the x-axis at points corresponding to its real solutions.However, when dealing with quadratic equations like \(x^2 = -18\), where we have imaginary solutions, the parabola remains above (or below) the x-axis since the solutions do not touch the axis.Here's why it happens:

• The term \(-18\) means the parabola is shifted down by 18 units from where \(x^2 = 0\) would be.
• Without real x-intercepts, the parabola never crosses the x-axis.
• This gives insight that no real-number solutions exist, thus guiding us toward imaginary solutions.

Understanding the graphical representation of equations with imaginary solutions reinforces their difference from real-number solutions.

Complex Numbers

Complex numbers are a broad class of numbers that include both real and imaginary components. They are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In the solutions of the equation \(x^2 = -18\), once simplified, we arrive at \(x = \pm 3i\sqrt{2}\). In this form:

• The real part \(a\) is 0 as there's no real number.
• The imaginary part \(b\) is \(\pm 3\sqrt{2}\), associated with "\(i\)".
• These numbers are purely imaginary because their real component is zero.

Complex numbers, including imaginary numbers, extend real numbers. They are exceptionally useful in advanced mathematics, engineering, and physics, providing solutions and understanding in systems where real solutions are inadequate.