Problem 117 - Intermediate Algebra

Question

Simplify. Determine whether \(2 i\) is a solution of \(x^{2}+4=0\)

Step-by-Step Solution

Answer

Yes, \(2i\) is a solution of the equation \(x^2 + 4 = 0\).

Step 1: Understand the given equation

The problem asks us to determine if \(2i\) is a solution to the equation \(x^2 + 4 = 0\). This involves substituting \(2i\) into the equation and checking if the equation holds true.

Step 2: Substitute \(2i\) into the equation

Replace \(x\) with \(2i\) in the equation \(x^2 + 4 = 0\). This gives us \((2i)^2 + 4 = 0\).

Step 3: Calculate \((2i)^2\)

To simplify \((2i)^2\), apply the formula \((a \, b)^2 = a^2 + 2ab + b^2\). Here, it simplifies to \(4i^2\). Given that \(i^2 = -1\), we get \(4i^2 = 4(-1) = -4\).

Step 4: Simplify the equation

Substitute the value calculated in Step 3 back into the equation to get \(-4 + 4 = 0\).

Step 5: Verify if the equation holds true

Since \(-4 + 4 = 0\) is a true statement, this confirms that \(2i\) satisfies the equation \(x^2 + 4 = 0\). Hence, \(2i\) is a solution to the equation.

Key Concepts

Quadratic Equations

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The characteristic feature of a quadratic equation is the square term, \( x^2 \). These equations often have two solutions. This is because they can be seen on a graph as a parabola that may intersect the x-axis at two points. For our exercise, we are tasked with solving \( x^2 + 4 = 0 \), a quadratic equation with no linear term (that is, \( b = 0 \)).

To solve these equations, we can rearrange them to isolate the square term and solve for \( x \). If the quadratic equation involves imaginary numbers, as in this problem, it means one or both solutions could involve complex numbers. This leads us to our next key concept, the imaginary unit. Remember that not all quadratic equations have real-number solutions, and exploring complex numbers can provide a comprehensive understanding.

Imaginary Unit

The imaginary unit, denoted as \( i \), is defined by the property that \( i^2 = -1 \). Imaginary numbers are used in mathematics to extend our number system and solve equations that do not have real solutions. By using the imaginary unit, we can solve equations like \( x^2 + 4 = 0 \), which do not intersect the x-axis if plotted in the real number system.

In this exercise, substituting \( 2i \) into the equation illustrates the role of the imaginary unit. When you substitute and simplify, you find that \( (2i)^2 = 4i^2 = 4(-1) = -4 \). Adding this result to the constant term from our equation \( x^2 + 4 = 0 \), we have \(-4 + 4 = 0\), confirming \( 2i \) as a valid solution. When tackling problems involving the imaginary unit, always remember \( i^2 = -1 \) as a foundational rule. This conversion allows us to convert complex equations into manageable forms.

Equation Solving

Equation solving involves finding the values that satisfy a given mathematical statement. For quadratic equations, this means determining what values of \( x \) make the equation true. A major strategy involves substituting potential solutions into the original equation to verify their validity.

• The first step is to substitute the proposed solution into the equation.
• Next, perform algebraic manipulations to simplify the expression.
• Finally, check if the simplified expression resolves to a true statement (like \( 0 = 0 \)).

In our solved exercise, we checked if \( 2i \) was a solution by following these steps. After calculating \( (2i)^2 \), we found \( -4 + 4 = 0 \). Since this is true, we confirmed the solution's validity. This method is crucial not just for this specific problem but for solving various equations in algebra, ensuring you've appropriately tested potential solutions.