Problem 80 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$w^{2}=-225$$

Step-by-Step Solution

Answer

The solutions are \( w = 15i \) and \( w = -15i \).

Step 1: Understand the Equation

The given equation is a quadratic equation in the form of \( w^2 = -225 \). The goal is to find the values of \(w\) that satisfy this equation, where \( w \) could be real or imaginary numbers.

Step 2: Isolate the Square Term

In the equation \( w^2 = -225 \), the square term \( w^2 \) is already isolated on one side of the equation.

Step 3: Take the Square Root of Both Sides

To solve for \( w \), take the square root of both sides:\( w = \sqrt{-225} \). Remember that the square root of a negative number involves an imaginary unit \( i \).

Step 4: Simplify the Square Root

Separate the square root of the negative number into the product of the square root of 225 and the imaginary unit \( i \).\( w = \sqrt{225} \sqrt{-1} \). Since \sqrt{-1} = i, you get \( w = \sqrt{225} i \).

Step 5: Evaluate the Square Root

The square root of 225 is 15. Therefore, \( w = 15i \). Since square roots have both positive and negative solutions, the final solutions are \( w = 15i \) and \( w = -15i \).

Key Concepts

quadratic equations

A quadratic equation involves terms up to the second power and follows the general form \[ax^2 + bx + c = 0\].
In the exercise, we have a simpler case without the linear and constant terms, i.e., \[w^2 + 0w - 225 = 0\]. These equations often have two solutions.
To solve, you can use various methods, such as factoring, completing the square, or using the quadratic formula.
In this exercise, isolating the square term and taking the square root offers a straightforward approach.
The solutions could be either real or imaginary, depending on the equation's discriminant (which can be positive, zero, or negative).

imaginary numbers

Imaginary numbers emerge when dealing with the square root of negative numbers.
In the context of our quadratic equation \[w^2 = -225\], to find the value of \(w\), we need to take the square root of \(-225\), which introduces the imaginary unit \(i\).
The imaginary unit \(i\) is defined as \(\root{-1}\).
This makes \(\root{-225}\) equal to \(\root{225}i\).
Understanding that \(w = \root{225}i\) is crucial, and solving further will provide both positive and negative solutions, hence \(w = \root{225}i\) or \(w = -\root{225}i\).
This shows that our final answers are \(w = 15i\) and \(w = -15i\).

square roots

The concept of square roots involves finding a number that, when multiplied by itself, yields the original number.
Square roots have two solutions: a positive and a negative counterpart.
In our equation \(w^2 = -225\), by isolating \(w\) on one side, we then take the square root of both sides to solve for \(w\).
To handle \(\root{-225}\), first, factor it as \(\root{225} \root{-1}\).
The square root of \(225\) is \(15\), and the square root of \(-1\) is \(i\).
So, we get \(w = 15i\), and since square roots involve both positive and negative values, we end up with \(w = 15i\) and \(w = -15i\).