Problem 16 - Precalculus : Building Concepts and Connections

Question

Use the definition of i to solve the equation. $$x^{2}=-25$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=-25\) are \(x=5i\) and \(x=-5i\).

Step 1: Identify type of equation

Recognize the equation \(x^{2}=-25\) as a quadratic equation.

Step 2: Introduce the imaginary unit

The square root of a negative number is not defined within the system of real numbers. Therefore, this implies we will have a solution in the set of imaginary numbers. This is possible by introducing the imaginary unit \(i\) where \(i^{2}=-1\). By definition, \(i=\sqrt{-1}\) . So we rewrite the equation as \(x^{2}=25*(i^{2})\).

Step 3: Use square root property

The square root property states that if \(x^{2}=a\), the solution is \(x=\sqrt{a}\) and \(x=-\sqrt{a}\). Applying this property, we have that \(x=\sqrt{25*(i^{2})}\) and \(x=-\sqrt{25*(i^{2})}\).

Step 4: Compute solution

Solve for \(x\). We have that \(x=5i\) and \(x=-5i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the realm of complex numbers. The introduction of \(i\) helps us handle the square roots of negative numbers, which are not possible within the real number system. By definition, \(i\) is such that \(i^2 = -1\). This imaginary unit allows us to extend our number system to include numbers that are not real, often called "imaginary numbers."

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)

With this, we can express the square root of any negative number \(\sqrt{-a}\) as \(i\sqrt{a}\), where \(a\) is a positive real number. Thus, the expression \(\sqrt{-25}\) can be rewritten as \(5i\). This form of representation is essential when working with complex equations that involve negative square roots.

Quadratic Equation

A quadratic equation is one of the most common types of polynomial equations. It typically takes the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). In the original exercise, the equation is \(x^2 = -25\), which is a simplified quadratic equation.
Quadratic equations can have two, one, or no real solutions depending on the discriminant \((b^2 - 4ac)\). When the discriminant is negative, the solutions are not real numbers; instead, they involve the imaginary unit \(i\). In such cases, the solutions manifest as complex numbers, indicating that every quadratic equation has two solutions in the realm of complex numbers.
Some key points related to quadratic equations include:

• They can have up to two solutions.
• Solutions can be real or complex according to the discriminant.
• When the equation involves negatives under a square root, the imaginary unit is involved.

Square Root Property

The square root property is a mathematical concept used to solve quadratic equations of the form \(x^2 = a\). According to this property:

• If \(x^2 = a\), then \(x = \sqrt{a}\) or \(x = -\sqrt{a}\).

This means that any quadratic equation of the form \(x^2 - a = 0\) can be solved by taking the square root of both sides. When dealing with negative numbers, as in \(x^2 = -25\), the square root property still applies, but it involves the imaginary unit \(i\). The solutions become \(x = \sqrt{-25} = 5i\) and \(x = -\sqrt{-25} = -5i\). Thus, applying the square root property naturally leads to solutions in the complex plane when negatives are involved under the square root.