Problem 1 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(x^{2}+4=0\)

Step-by-Step Solution

Answer

Two complex roots: \(x = 2i\) and \(x = -2i\).

Step 1: Understand the Equation Type

The given equation is a quadratic equation of the form \(x^2 + 4 = 0\). Quadratic equations are generally in the form \(ax^2 + bx + c = 0\). This equation has \(a = 1\), \(b = 0\), and \(c = 4\).

Step 2: Set Equation for Solving

To solve \(x^2 + 4 = 0\), we start by isolating \(x^2\) on one side: \(x^2 = -4\).

Step 3: Solve for x Using Square Roots

To solve \(x^2 = -4\), take the square root of both sides. This gives \(x = \pm\sqrt{-4}\).

Step 4: Simplify the Square Root

The square root of \(-4\) can be expressed in terms of imaginary numbers. We know \(\sqrt{-1} = i\), so \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).

Step 5: State the Solutions

Thus, the solutions for the equation \(x^2 + 4 = 0\) are \(x = 2i\) and \(x = -2i\).

Step 6: Determine the Number and Type of Roots

There are two roots, both of which are complex (non-real) and imaginary.

Key Concepts

Imaginary Numbers

In mathematics, numbers can be classified into different types based on their properties and how they behave under operations. Real numbers include all the "usual" numbers you work with in everyday situations, such as whole numbers, fractions, and decimals.
However, you might encounter situations, like with the equation in this exercise, where you need the concept of imaginary numbers. Imaginary numbers arise when you try to find the square root of a negative number, which is not possible within the realm of real numbers.
The key to understanding imaginary numbers is the imaginary unit, denoted by "i". It is defined such that:

• i = \(\sqrt{-1}\)
• The property \(i^2 = -1\)

Using these properties, any negative square root can be rewritten as a real number multiplied by "i". For example, the square root of -4 becomes \(2i\), because \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i\). This simple transformation allows us to extend our number system and solve equations that would otherwise be unsolvable.

Complex Roots

When solving quadratic equations, it's important to determine the nature of the roots. The roots of a quadratic equation can be real or complex. Complex roots occur when the discriminant, that is \(b^2 - 4ac\), in the quadratic formula is less than zero.

For a quadratic equation like \(x^2 + 4 = 0\), let's identify the terms:

• a = 1
• b = 0
• c = 4

Substituting these, the discriminant \( b^2 - 4ac \) equates to \(0 - 16\) or \(-16\). Since the discriminant is negative, the equation has complex roots.
These complex roots arise from the necessity to take the square root of a negative number, leading to imaginary numbers. Since complex roots occur in conjugate pairs, for \(x^2 + 4 = 0\) they are \(x = 2i\) and \(x = -2i\). This ensures even complex solutions balance the equation.

Solving Quadratic Equations

A quadratic equation is typically given in the standard form: \(ax^2 + bx + c = 0\). However, solving these equations varies slightly based on the nature of the equation's roots. Here's how you approach solving a quadratic equation:

• Identify the equation's coefficients \(a\), \(b\), and \(c\).
• Determine if it can be factored, or if a formulaic approach is necessary (such as the quadratic formula or completing the square).

For the equation \(x^2 + 4 = 0\), we see quickly it cannot be factored over real numbers, so we isolate \(x^2\) giving \(x^2 = -4\).
Taking the square root of both sides yields complex roots, involving imaginary numbers. To solve, recognize \(\sqrt{-4}\) as \(2i\) and therefore solve \(x = 2i\) and \(x = -2i\).
In summary, understanding both real and imaginary components, as well as the use of properties such as \(i\), is crucial in efficiently solving quadratic equations with complex solutions.