Problem 91 - Algebra A Combined Function

Question

Without solving, determine whether the solutions of each equation are real numbers or complex, but not real numbers. See the Concept Check in this section. $$ (2 y-5)^{2}+7=3 $$

Step-by-Step Solution

Answer

The solutions are complex numbers, not real.

Step 1: Understand the Equation Structure

The given equation is \((2y-5)^2 + 7 = 3\). Our task is to determine if the solutions are real or complex without solving it.

Step 2: Rearrange the Equation

First, let's isolate the squared term by subtracting 7 from both sides: \((2y-5)^2 = 3 - 7\). This simplifies to \((2y-5)^2 = -4\).

Step 3: Analyze the Nature of the Squared Term

A squared term \((2y-5)^2\) equals \(-4\). Typically, a squared term cannot be negative, since squaring a real number cannot yield a negative result. This implies the value inside the square must be complex in order to satisfy the equation.

Step 4: Conclusion on the Nature of the Solution

Given that \((2y-5)^2 = -4\), the squared term cannot result in a negative value if \(y\) is real. Therefore, the equation will have complex solutions, specifically involving imaginary numbers.

Key Concepts

Imaginary Numbers

Imaginary numbers might seem like magical concepts at first, yet they are quite fundamental in mathematics. These numbers allow us to solve equations that have no real-number solutions. One key idea to grasp is that an imaginary number, often denoted as \(i\), is defined such that \(i^2 = -1\). This stands in contrast to real numbers, where squaring any real number results in a non-negative value.

In the equation from the exercise, you see that the squared term \((2y-5)^2\) equals \(-4\). In standard real-number arithmetic, squaring a term never results in a negative number, underscoring that part of it must involve imaginary entities. Thus, when solving such equations, imaginary numbers come into play, simplifying expressions like \(-4\) into \(4i^2\).

Understanding imaginary numbers is key when navigating equations involving negative results from squared terms. They open new pathways to solutions that otherwise seem out of reach in real-number contexts.

Real Numbers

Real numbers form the backbone of everyday arithmetic and basic mathematics. They include all the positive and negative integers, fractions, and irrational numbers like \(\sqrt{2}\) and \(\pi\). These numbers can be depicted on a number line extending infinitely in both the positive and negative directions.

When dealing with equations, we often first check for real solutions, focusing on whether the result of operations like squaring, adding, or subtracting yields values that are feasible within the realm of real numbers. In the context of the exercise given, we explore whether the operation \((2y-5)^2\) resulting in \(-4\) can reside within real numbers.

Given that squaring any real number always results in a non-negative value, the equation's notion of \((2y-5)^2 = -4\) falls outside the domain of real numbers. This means our solutions must involve more than just these traditional figures, entertaining the possibilities of imaginary numbers to fully comprehend the result.

Equation Analysis

Equations are the mathematician's tool to express relationships, predict outcomes, and identify unknowns. Analyzing equations involves understanding their structure to infer the nature of their solutions, whether real or complex.

The example equation \((2y-5)^2 + 7 = 3\) demonstrates a typical scenario where we inspect the form to determine solution types without solving it entirely. We first rearrange the equation to isolate terms, such as obtaining \((2y-5)^2 = -4\) from moving constants across the equation. This rearrangement helps us observe potential solution characteristics.

Part of equation analysis is recognizing unusual scenarios where results deviate from expected real-number outputs, like the negative result of a squared term here. By identifying such aspects, we understand that the equation likely offers complex solutions necessitating imaginary numbers. Hence, equation analysis helps us to categorize and solve problems accurately, anticipating their behavior with various mathematical tools.