Problem 26 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$-25$$

Step-by-Step Solution

Answer

The square roots of -25 are \(5i\) and \(-5i\).

Step 1: Identify the number

The number we need to find the square root of is -25.

Step 2: Calculate the square root

We know that the square root of a negative number results in a complex number, so first, find the square root of 25, which is 5. Since the number is negative, the square root is a complex number, and can be either positive or negative, giving us the two roots: \(5i\) and \(-5i\). These are the two numbers that when squared give \(-25\).

Step 3: Check the results

To ensure that \(5i\) and \(-5i\) are indeed the roots of \(-25\), square both numbers. Squaring \(5i\) gives us \((5i)^2 = 25i^2 = 25*(-1) = -25\). Squaring \(-5i\) gives us \((-5i)^2 = 25i^2 = 25*(-1) = -25\). Therefore the roots are correct, as both \(5i\) and \(-5i\) when squared result in \(-25\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. Typically, they are expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of \(-1\).

• The real number part, \(a\), is any number from the set of real numbers.
• The imaginary part involves \(i\), turning the number "complex."

Thus, when we tackle a negative square root, such as \(-25\), we enter the realm of complex numbers because we need \(i\) to express the square root. For \(-25\), we first consider the square root of the positive part, 25, which is 5. Then, we multiply by \(i\), resulting in complex numbers \(5i\) and \(-5i\). These demonstrate how complex numbers elegantly handle the square roots of negative numbers.

Imaginary Numbers

An imaginary number represents a number whose square is negative. Every imaginary number can be written as \(bi\) where \(b\) is a real number, and \(i\) is the imaginary unit. Key things to remember about imaginary numbers include:

• The imaginary unit \(i\) squares to give \(-1\), i.e., \(i^2 = -1\).
• Imaginary numbers help solve equations that cannot be solved using real numbers alone.

In the problem of finding square roots for \(-25\), the imaginary component \(i\) becomes essential. It enables us to express the solutions as \(5i\) and \(-5i\). Without imaginary numbers, square roots of negative numbers would remain elusive and unsolvable using traditional real numbers.

Verifying Square Roots

Verifying if a number is a square root involves checking if squaring the number returns the original value. For complex numbers, this process holds the same importance. Let's consider both roots, \(5i\) and \(-5i\), for the number \(-25\):

• Calculate \((5i)^2\): - the formula \((a)^2\) means multiplying \(a\) by itself. You'll get \(5i \times 5i = 25i^2 = 25(-1) = -25\).
• Similarly, \((-5i)^2\) also follows the same pattern resulting in \(-25\).

This step is crucial as it confirms the accuracy of the square roots. In our case, both \(5i\) and \(-5i\) accurately return \(-25\) upon squaring, confirming they are indeed the correct square roots. Through squaring, we ensure no miscalculations occurred while finding the roots.