Problem 92 - 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. $$ (3 m+2)^{2}+4=1 $$

Step-by-Step Solution

Answer

The solutions are complex numbers, not real numbers.

Step 1: Understand the Structure of the Equation

Observe that the equation \((3m+2)^2 + 4 = 1\) is a quadratic equation in disguised form because it involves the square of a binomial.

Step 2: Isolate the Squared Term

To analyze the equation, move 4 to the right side. This gives us \((3m + 2)^2 = 1 - 4\). Simplifying the right side: \((3m + 2)^2 = -3\).

Step 3: Determine the Nature of the Equation

The equation \((3m + 2)^2 = -3\) involves squaring a term to equal a negative number. A real number squared can never be negative. Thus, this equation has no real solutions.

Step 4: Identify the Type of Solutions

Since the square of a real number cannot be negative, the solutions to the equation must be complex numbers, based on the squared term being equal to a negative value.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. Usually, it takes the general form of \( ax^2 + bx + c = 0 \). The equation in the problem looks different but is still a quadratic equation because it involves the square of a binomial: \((3m + 2)^2 + 4 = 1\). It's essential to understand how to identify quadratic equations even when they appear in a disguised form.To work with these, one approach is to rewrite them in the standard form by isolating the squared term, as seen in the step-by-step solution. The reshaping of the equation into \((3m + 2)^2 = -3\) highlights the quadratic nature, where the expression inside the square is set to a constant.

Real Numbers

Real numbers make up a number set that includes all the integers, fractions, and irrational numbers, such as \(\pi\) and \(\sqrt{2} \). They represent all the points on an infinitely long number line. However, when solving equations, especially quadratics, it's crucial to determine if solutions are real.The pivotal point in this problem is understanding that when we encounter the equation \((3m + 2)^2 = -3\), it implies searching within the real numbers to see if there is a solution. Here, we find a negative on the right-hand side which is impossible when dealing with real numbers, since no real number squared results in a negative value. This diagnostic is key in understanding that the lack of real solutions implies the solutions must be complex.

Binomial

A binomial is a type of polynomial that consists of two terms, often expressed in the form \(a + b\) or \(a - b\). Quadratic equations frequently involve a binomial squared, as in this problem's equation, \((3m + 2)^2\).Grasping this structure is essential since the solution involves manipulating these binomials through algebraic techniques. Since the square of \((3m+2)\) results in an expression like \((3m+2)^2=9m^2 + 12m + 4\), it's vital to understand how the binomial configuration plays a role in constructing a quadratic equation. Recognizing the binomial term makes identifying the form of an equation clearer, aiding in deciding whether real or complex solutions are expected.