Problem 25 - College Algebra Essentials

Question

Solve each equation in Exercises \(15-34\) by the square root property. $$ (x+3)^{2}=-16 $$

Step-by-Step Solution

Answer

The solutions to the equation \((x+3)^{2}=-16\) are \(x = -3 + 4i\) and \(x = -3 - 4i\).

Step 1: Isolate the Squared Term

Arrange the equation so that the squared term is alone on one side. For the initial equation \((x+3)^{2}=-16\), this is already the case.

Step 2: Apply Square Root Property

The next step is to use the square root property. However, the expression \((x+3)^{2}=-16\) involves the square root of a negative number. In this case, we rewrite the negative number as \(-16 = 16*(-1)\), which allows us to introduce the imaginary unit \(i\) (where \(i^{2}=-1\)). Taking the square root of both sides of the equation we obtain \(x+3 = \pm \sqrt{16*(-1)} = \pm 4i\).

Step 3: Find Values of x

Lastly, isolate \(x\) to find the solutions. Subtracting 3 from both sides gives \(x = -3 \pm 4i\).

Key Concepts

Square Root Property

The square root property is a helpful tool in solving equations, particularly those involving squares. It states that if you have an equation where a variable squared equals a number, you can take the square root of both sides to solve for the variable.

Here's how it works: for an equation in the form \(x^2 = a\), we can take the square roots of both sides, resulting in \(x = \pm \sqrt{a}\). The plus-minus sign indicates that both the positive and negative roots of the number are solutions.

• This property is particularly useful for perfect squares and equations where the variable is set free to balance the equation.
• However, when the number on the other side of the equation is negative, as in this exercise, extra considerations, like complex numbers, are required.

Integrating these aspects ensures a comprehensive approach to applying the square root property in various scenarios.

Imaginary Unit

In mathematics, the imaginary unit, denoted as \(i\), is a fundamental concept necessary for dealing with negative square roots. By definition, \(i^2 = -1\). When solving equations that require taking the square root of a negative number, \(i\) becomes essential.

For example, in the provided equation, \((x+3)^2 = -16\), we need to take the square root of \(-16\). Here, \(-16 = 16 \times (-1)\). Therefore, \(\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\).

• \(i\) allows us to express the solution in terms of complex numbers.
• It extends our number system beyond real numbers, offering solutions when traditional real number methods fall short.

When the solutions to the equations involve \(i\), they become complex numbers, a crucial concept for understanding deeper mathematical ideas.

Solving Equations

Solving equations generally involves finding values for variables that make the equation true. When dealing with equations that involve complex numbers, like in this exercise, a systematic approach ensures accurate solutions.

The given equation is \((x+3)^2 = -16\). Follow these steps to solve equations of this nature:

• First, ensure the squared term is isolated. If not yet isolated, rearrange the equation accordingly.
• Apply the square root property to both sides of the equation, keeping in mind the introduction of the imaginary unit \(i\) when negative square roots are present.
• Finally, find the values of \(x\) by isolating them further, resulting in solutions in the form of complex numbers.

In this example, we have \(x+3 = \pm 4i\), leading us to the solutions \(x = -3 \pm 4i\). Complex solutions highlight the equation-solving process beyond simple arithmetic, ensuring a deeper understanding of mathematical principles.