Problem 44 - Algebra for College Students

Question

What is the graph of the equation \(x^{2}+y^{2}=-4\) ? \(\mathrm{Ex}-\) plain your answer.

Step-by-Step Solution

Answer

The equation doesn't have a graph, as squares cannot equal a negative number.

Step 1: Understand the Equation

The equation given is \(x^2 + y^2 = -4\). Generally, equations of the form \(x^2 + y^2 = r^2\) represent a circle with radius \(r\), centered at the origin.

Step 2: Analyze the Right Side of the Equation

In the equation \(x^2 + y^2 = -4\), the right side is \(-4\), which is negative. For a circle equation \(x^2 + y^2 = r^2\), \(r^2\) must be positive since the radius \(r\) is always a non-negative real number.

Step 3: Conclude on the Possibility of Graphing

Since \(-4\) is negative, the expression on the left side \(x^2 + y^2\) (which is the sum of squares) cannot equal a negative number because squares of real numbers are non-negative.

Step 4: State the Outcome

The equation \(x^2 + y^2 = -4\) does not represent any graph in the real number coordinate system. Therefore, this equation does not have a graph.

Key Concepts

Graphing Equations and Understanding Circles

Graphing equations often involves plotting mathematical terms on a coordinate plane. The specific equation format discussed here is typically one for a circle: \(x^2 + y^2 = r^2\). This equation suggests a circle with a radius \(r\) and centered at the origin of the coordinate system, designated by \((0,0)\).

• The radius \(r\) is calculated as the square root of the number on the right side (\(r^2\)).
• If \(r^2\) is positive, graphing results in a circle.
• If \(r^2\) is zero, it represents a single point; the circle's center.
• If \(r^2\) is negative, a circle cannot exist in the real number plane, as positive square results are required.

Understanding this helps identify when an equation can or cannot be graphed in familiar settings like the Cartesian plane. For instance, an equation like \(x^2 + y^2 = -4\) doesn't describe a real shape because you can't have a negative radius squared.

Exploring the Real Number System

The real number system is foundational in mathematics and consists of all numbers that can be found on the number line. This includes both rational numbers (like fractions and whole numbers) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).

Within this system, there are important properties to know:

• Squares of real numbers are always non-negative, as seen in calculations like \(x^2\) or \(y^2\).
• Negative results require extension beyond real numbers, entering the complex number domain.

In context, if the squared terms must total a negative number—as in the equation \(x^2 + y^2 = -4\)—it is an impossibility within the real number system, thereby confirming no valid graph exists.

Coordinate Geometry Basics

Coordinate geometry, also known as analytic geometry, involves describing geometry through algebraic equations. It enables the comparison of geometric figures using coordinates on the Cartesian plane.

Key Points About Coordinate Geometry:

• The coordinate plane has two axes: horizontal (\(x\)-axis) and vertical (\(y\)-axis).
• Points are defined by pairs, \((x, y)\), where \(x\) represents horizontal distance from origin and \(y\) represents vertical distance.
• Circular equations, such as \(x^2 + y^2 = r^2\), use these coordinates to determine the positions and shapes of circles.

In examining the equation \(x^2 + y^2 = -4\), coordinate geometry reveals inconsistency when trying to plot, due to the square terms collectively needing to equate to a negative, which naturally is undefined for real values.