Problem 79 - Beginning and Intermediate Algebra

Question

Solve each equation. (All solutions are nonreal complex numbers.) $$ (x+3)^{2}=-4 $$

Step-by-Step Solution

Answer

The solutions are \( x = -3 + 2i \) and \( x = -3 - 2i \).

Step 1: Recognize the Form of the Equation

Notice the equation \( (x+3)^{2} = -4 \) is a quadratic equation where the right-hand side is a negative number. This will result in nonreal complex solutions.

Step 2: Take the Square Root of Both Sides

Isolate \( (x+3) \) by taking the square root of both sides of the equation. \[ \sqrt{(x+3)^{2}} = \pm \sqrt{-4} \] This simplifies to \( x+3 = \pm 2i \) because \( \sqrt{-4} = \pm 2i \).

Step 3: Solve for x

Isolate x by subtracting 3 from both sides of each equation, resulting in: \[ x = -3 + 2i \quad \text{and} \quad x = -3 - 2i \]

Key Concepts

Nonreal Solutions

In mathematics, equations sometimes don't have real number solutions. Instead, they have solutions known as nonreal or complex solutions. A nonreal solution typically involves the term 'i,' the imaginary unit, which is defined as the square root of -1. Nonreal solutions arise especially when solving quadratic equations where the discriminant (the part under the square root in the quadratic formula) is negative. Recognizing nonreal solutions is important because they help us solve equations that otherwise would have no answer in the set of real numbers.

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They are solved by factoring, completing the square, or using the quadratic formula. In this exercise, the equation given \( (x+3)^2 = -4 \) is already in a special form. When you see a quadratic equation like this, your goal is to isolate the x-variable by performing algebraic operations.

Complex Numbers

Complex numbers expand our understanding beyond real numbers. A complex number is in the form \(a + bi \), where \('a' is the real part and 'b' is the imaginary part, with i being \sqrt{-1}\). For example, in our solution, \((-3 + 2i)\) and \((-3 - 2i)\) are complex numbers. The imaginary component \('2i'\) arises because we took the square root of a negative number, which doesn't exist among real numbers.

Square Roots

Taking the square root of both sides is an essential step when solving quadratic equations like \( (x+3)^2 = -4 \). The square root of a negative number introduces the imaginary unit 'i.' For instance, \(\sqrt{-4} = \pm 2i\) because \(\sqrt{-1} = i\). By understanding this, you simplify the equation to isolate the variable x. This method helps us solve equations that have no real solutions but instead have nonreal complex ones.