Problem 44 - Algebra 2

Question

Solve each equation. $$ 2 x^{2}+12=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \sqrt{6}i \) and \( x = -\sqrt{6}i \).

Step 1: Move Constant to the Other Side

We begin by isolating the term with the variable. Subtract 12 from both sides of the equation: \[ 2x^2 + 12 - 12 = 0 - 12 \]which simplifies to:\[ 2x^2 = -12 \]

Step 2: Divide by the Coefficient of x^2

Divide each term on both sides of the equation by 2 to solve for \(x^2\): \[ \frac{2x^2}{2} = \frac{-12}{2} \]which simplifies to:\[ x^2 = -6 \]

Step 3: Apply the Square Root

Take the square root of both sides to solve for \(x\), considering both positive and negative roots:\[ x = \pm \sqrt{-6} \]

Step 4: Simplify Using Imaginary Numbers

Since \( \sqrt{-6} \) involves the square root of a negative number, it can be rewritten using imaginary numbers:\[ x = \pm \sqrt{6}i \]

Key Concepts

Imaginary Numbers

When solving quadratic equations such as the one in our problem, you might encounter the square root of a negative number. Typically, real numbers cannot accommodate the square root of negatives, and that's where imaginary numbers come into play. The core idea of imaginary numbers revolves around the imaginary unit denoted as "i."

• The imaginary unit "i" is defined by the property that \( i^2 = -1 \).
• This means that \( \sqrt{-1} = i \).

Once we understand and accept this concept, we can extend it to express the square root of any negative number. For instance, \( \sqrt{-6} \) can be expressed as \( \sqrt{6}i \). By using imaginary numbers, we can handle equations that have no real solution, making it possible to solve a wider variety of mathematical problems.

Square Root Method

The square root method is a straightforward technique to solve quadratic equations when they are set in the form \( ax^2 = c \). It involves directly taking the square root of both sides of the equation. Here’s a simple breakdown of this method:

• Rearrange: First, ensure that the quadratic equation is in the form \( x^2 = d \). For our problem, it becomes \( x^2 = -6 \) after moving terms around correctly.
• Square Root: Apply the square root to both sides. Remember, you must consider both the positive and negative roots. This is because squaring either a positive or negative number results in a positive number.
• Include i: When the square root involves a negative value, incorporate "i" to obtain the correct form of the solution.

For example, solving \( x^2 = 4 \) will yield \( x = \pm 2 \). Similarly, \( x^2 = -6 \) gives us \( x = \pm \sqrt{6}i \) after applying the concept of imaginary numbers.

Complex Solutions

Whenever a quadratic equation's solutions include imaginary numbers, these solutions are referred to as complex solutions. They have a real part and an imaginary part, typically written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

• For the given problem, the solution was \( x = \pm \sqrt{6}i \), which is actually a pure imaginary number.
• Complex solutions often arise when the quadratic formula or other methods lead to a square root of a negative number.

It's important to note that complex solutions always come in conjugate pairs. This symmetry is crucial because if \( x_1 = a + bi \) is a solution, then \( x_2 = a - bi \) will also be a solution due to the nature of quadratic equations. Both solutions together give a full understanding of how the parabola associated with the quadratic equation intersects the complex plane, not just the real number line. This concept expands the possibilities and applications of solving quadratic equations beyond just the realm of real numbers.