Problem 19 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation \(x^2 = -18\) has no real solutions, only complex solutions: \(x = \pm 3i\sqrt{2}\).

Step 1: Analyze the Equation

The given equation is \(x^2 = -18\). This is a quadratic equation, but notice that the right side is negative. In the real number system, there are no real numbers whose square is negative.

Step 2: Consider the Solution Set

Since no real number squared equals -18, we conclude that there are no real solutions for the equation \(x^2 = -18\). However, there can be complex solutions.

Step 3: Find Complex Solutions

To find complex solutions, equate \(x^2 = -18\) to \(x = \pm i\sqrt{18}\). This simplifies to \(x = \pm 3i\sqrt{2}\), where \(i\) is the imaginary unit and \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).

Step 4: Graphical Interpretation

Graphically, the parabola \(y = x^2\) lies entirely above the x-axis for real \(x\), crossing the y-axis at point (0,0). Since the graph of \(x^2\) does not intersect the line \(y = -18\) on the real plane, there are no real intersections, which means no real solutions.

Key Concepts

Understanding Complex Solutions

When we talk about complex solutions in quadratic equations, we enter the realm of numbers that are not part of our usual real number system. The equation \(x^2 = -18\) is a prime example of this. In the real number system, there is no real number whose square results in a negative. Hence, quoting from above, we realize that real solutions are impossible.

• Complex numbers introduce an entity called the imaginary unit denoted by \(i\), which is defined as \(i^2 = -1\).
• Using this, we can express \(i\sqrt{18}\), where \(\sqrt{18} = 3\sqrt{2}\), allowing us to solve our equation as \(x = \pm 3i\sqrt{2}\).
• Thus, the solutions are complex numbers \(x = 3i\sqrt{2}\) and \(x = -3i\sqrt{2}\).

These complex numbers help expand solutions beyond the typical real number answers, letting us "solve" equations that initially seem unsolvable.

Exploring the Real Number System

The real number system includes all the numbers we deal with daily—integers, fractions, and irrationals such as \(\pi\) and \(\sqrt{2}\). All these are real because they can be found on a standard number line.

• Revisiting our equation \(x^2 = -18\), we notice we're limited by this system because real numbers can't satisfy this equation.
• The equation doesn't fit because squaring any real number yields a non-negative result, never negative.

Since \(x^2\) results in \(y\) being non-negative for all real \(x\), our equation lives outside the real number bounds, leading us into the domain of complex solutions.

Graphical Interpretation and Solution Visualization

Graphically interpreting quadratic equations involves plotting the parabola \(y = x^2\) and examining where it meets another line or curve. For \(x^2 = -18\), or equivalently, where \(y = -18\), we face an interesting scenario.

• The parabola \(y = x^2\) is entirely above the x-axis for real numbers, indicating all outputs \(y\) are non-negative.


• Since the line \(y = -18\) is well below the x-axis, there is no intersection between \(y = x^2\) and \(y = -18\) on a real plane graph.


• This lack of intersection confirms the absence of real solutions graphically, hence encouraging the exploring of complex solutions where imaginary domains help offer meaningful solutions instead.

Visualizing this interaction helps understand why quadratic equations sometimes must yield to complex solutions, providing a full picture of the equation's properties.