Problem 25 - Algebra A Combined Function

Question

Use the square root property to solve each equation. $$ (x+3)^{2}+8=0 $$

Step-by-Step Solution

Answer

No real solution.

Step 1: Isolate the squared term

Begin by subtracting 8 from both sides of the equation to isolate the square term. This results in the equation: \( (x+3)^2 = -8 \).

Step 2: Identify the impossibility

Recognize that you cannot take the square root of a negative number in the set of real numbers without involving imaginary numbers. This means there is no real solution to the equation.

Step 3: Confirm understanding with imaginary numbers (optional)

If you were to proceed considering imaginary numbers, you would take the square root of both sides: \( x + 3 = \pm i\sqrt{8} \). However, this requires understanding complex numbers, often not covered in basic algebra. Conclude that there are no real numbers that satisfy this equation.

Key Concepts

Isolating Squared Terms

When working with equations that involve squared expressions, such as \((x + 3)^2 + 8 = 0\), the first step in solving is isolating the squared term. This involves removing any constants or terms not directly attached to the square. Here’s how to do it:

• Subtract the constant on the same side as the squared term from both sides of the equation. In this case, subtract 8 from both sides to obtain \((x + 3)^2 = -8\).

This step is crucial to simplify the equation, making it easier to solve. By isolating the squared term, you can apply methods like taking the square root to solve for the variable inside the square.

Imaginary Numbers

Imaginary numbers come into play when dealing with the square root of negative numbers. In the standard number line of real numbers, squaring any number—whether positive or negative—results in a positive number or zero. Thus, the square root of a negative number doesn't exist in the real number realm.

• An imaginary number is defined as the square root of \(-1\) and is denoted as \(i\). Thus, \(i^2 = -1\).
• If you encounter a square root of a negative number, convert this into a multiple of \(i\). For example, the square root of \(-8\) can be expressed as \(i\sqrt{8}\).

Understanding imaginary numbers is important when solving equations that have no real solutions, as it allows us to extend our number system to include solutions in the form of complex numbers.

Complex Numbers

Complex numbers combine real and imaginary numbers into a single entity, represented as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The concept of complex numbers is essential in advanced algebra and beyond, particularly in solving polynomial equations that have no real solutions.

• In the equation \((x + 3)^2 = -8\), recognizing the need to use imaginary numbers leads to solutions in the form of complex numbers. Taking the square root of both sides gives \(x + 3 = \pm i\sqrt{8}\).
• To simplify, solve for \(x\) by subtracting 3 from both sides: \(x = -3 \pm i\sqrt{8}\).

Though complex numbers might seem abstract, they are incredibly useful in fields such as engineering and physics, providing solutions where real numbers fall short.