Problem 11 - Algebra 2

Question

Solve each equation. $$ 2 x^{2}+18=0 $$

Step-by-Step Solution

Answer

The solutions are \(3i\) and \(-3i\).

Step 1: Move the Constant Term

Firstly, subtract 18 from both sides of the equation to isolate the term with the variable.\[2x^2 = -18\]

Step 2: Divide by the Coefficient

To simplify the equation, divide both sides by 2.\[x^2 = -9\]

Step 3: Take the Square Root

Take the square root of both sides. Don't forget that taking the square root of a negative number introduces the imaginary unit \(i\), where \(i = \sqrt{-1}\).\[x = \pm \sqrt{-9} = \pm 3i\]

Step 4: Write the Solutions

The solutions to the equation are the complex numbers.\[x = 3i \quad \text{or} \quad x = -3i\]

Key Concepts

Imaginary Unit

In mathematics, particularly in the field of complex numbers, we encounter the concept of the imaginary unit, denoted by \( i \). The imaginary unit is defined as the square root of \( -1 \). This concept might seem a bit abstract at first, but it becomes very useful, especially when dealing with complex numbers.

• The imaginary unit \( i \) allows us to express complex numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
• It helps in solving equations that do not have real solutions. For instance, the equation \( x^2 + 1 = 0 \) can't be solved using real numbers. However, using the imaginary unit, the solution is \( x = \pm i \).

Remember, \( i^2 = -1 \), which is a fundamental property used in various mathematical operations involving imaginary numbers.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \). These equations appear frequently in algebra and have different methods for solving them, such as factoring, completing the square, or using the quadratic formula.

• The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), provides the solutions for any quadratic equation.
• The discriminant, \( b^2 - 4ac \), within the quadratic formula helps to determine the nature of the roots. If it's negative, the solutions are complex numbers.

The given exercise is an example of a quadratic equation where the solution involves complex numbers, illustrating the versatility and scope of quadratic equations.

Square Root of Negative Number

The idea of finding the square root of a negative number is closely tied to the introduction of the imaginary unit. Normally, the square root function is defined only for non-negative values in the real number system. But, the concept of the square root of negative numbers opens up the realm of complex numbers.

• To compute the square root of a negative number, such as \( -9 \), we break it down to \( \sqrt{9} \times \sqrt{-1} \). This yields \( 3i \), using the property that \( i = \sqrt{-1} \).
• Complex solutions like \( x = 3i \) and \( x = -3i \) involve considering both the positive and negative roots, just as we do with positive numbers.

The exercise shows how taking the square root of a negative number directly brings us into the territory of imaginary numbers, turning negative roots into valid complex solutions.