Problem 28 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (x+2)^{2}=-7 $$

Step-by-Step Solution

Answer

The solutions to the equation, in the given context, are imaginary numbers and can be written as: \(x = -2 ± i \sqrt{7}\)

Step 1: Apply Square Root

Firstly apply the square root property on both sides of the equation. Remember that when rooting both sides of an equation, it will yield positive and negative results. So the operation will look as follows: \(\sqrt{(x+2)^{2}} = \sqrt{-7}\).

Step 2: Evaluate Square Root

After the application of the square root property, the equation looks like this: \(x + 2 = ± \sqrt{-7}\). However, the square root of a negative number is an imaginary number. Thus, we use the principle of imaginary numbers to rewrite \(\sqrt{-7}\) as \(i \sqrt{7}\).

Step 3: Simplify the Equation

At this point, having addressed the presence of imaginary numbers, we subtract 2 from both sides of the equation to isolate x. This gives us the final solution: \(x = -2 ± i \sqrt{7}\).

Key Concepts

Imaginary Numbers

When solving equations, you may encounter the square root of a negative number. This is where imaginary numbers come into play. Imaginary numbers are essential for understanding solutions that don't fit within the realm of real numbers.
Imaginary numbers are defined as multiples of the imaginary unit, denoted by \(i\), which is the square root of -1. Hence, \(i^2 = -1\). When you encounter the square root of any negative number, you can express it using \(i\). For example, \(\sqrt{-7}\) becomes \(i \sqrt{7}\).
Imaginary numbers form the basis of complex numbers, which allow solutions to be expressed in a broader sense, beyond the limits of real numbers.

Complex Solutions

A complex solution consists of both a real and an imaginary component. Understanding this concept is crucial when dealing with equations like \((x + 2)^2 = -7\). The imaginary part arises when you take the square root of a negative number, and the real part is derived from the manipulations of the equation.
Complex solutions are written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. In our exercise, once resolved, the solution is expressed as \(x = -2 \pm i \sqrt{7}\). Here, \(-2\) is the real part and \(\pm i \sqrt{7}\) is the imaginary part.

• Such solutions tell us that the equation does not have real roots.
• They are especially useful in fields like engineering and physics.

They offer a more complete picture of potential solutions, even if we can't graph them on the usual number line.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. The square root property is one method to solve such equations, particularly when reducing them to perfect squares.
With the square root property, you simplify the equation by taking the square root of both sides. This introduces positive and negative possible outcomes. For example, applying this to \((x + 2)^2 = -7\) results in \(x + 2 = \pm \sqrt{-7}\).
Quadratic equations can have:

• Two real solutions
• One real solution (a double root)
• Two complex solutions, as we see here

Every quadratic equation, whether it involves real or complex numbers, adheres to these principles. Understanding how to manipulate these equations not only aids in academic pursuits but also enhances problem-solving skills in practical applications.