Problem 66 - Algebra 2

Question

Solve each equation, and locate the complex solutions in the complex plane. $$ -3 x^{2}-9=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = i\sqrt{3}\) and \(x = -i\sqrt{3}\).

Step 1: Write the equation in standard form

The given equation is \(-3x^2 - 9 = 0\). First, we recognize the equation is already in standard quadratic form: \(ax^2 + bx + c = 0\) where \(a = -3\), \(b = 0\), and \(c = -9\).

Step 2: Isolate the term with the variable

To simplify the equation, first isolate the \(x^2\) term. Add \(9\) to both sides to get: \(-3x^2 = 9\).

Step 3: Solve for \(x^2\)

Divide both sides by \(-3\) to solve for \(x^2\):\[x^2 = \frac{9}{-3} = -3\]

Step 4: Solve for \(x\) by finding the square root

To find \(x\), take the square root of both sides. Remember, when taking the square root of a negative number, the result is a complex number:\[x = \pm \sqrt{-3} = \pm i\sqrt{3}\]

Step 5: Identify the roots in the complex plane

The solutions to the equation are \(x = i\sqrt{3}\) and \(x = -i\sqrt{3}\). In the complex plane, \(i\sqrt{3}\) lies on the positive imaginary axis at \(0 + i\sqrt{3}\), and \(-i\sqrt{3}\) lies on the negative imaginary axis at \(0 - i\sqrt{3}\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constant coefficients and \( x \) represents the variable we are solving for. Quadratic equations appear frequently in various mathematical contexts and practical problems, such as physics and engineering. Solving these equations allows us to find the values of \( x \) that make the equation true, known as the roots or solutions.

A quadratic equation can have different types of roots, depending on the discriminant \( b^2 - 4ac \):

• If positive, there are two distinct real roots.
• If zero, there is exactly one real root (a repeated root).
• If negative, the equation has two complex roots, which we'll explore further to understand how they are represented and processed mathematically.

Recognizing these distinct cases is essential in determining the methods used to find the solutions and where they stand on the real number line or in the complex plane.

Complex Plane

Complex numbers are not as abstract as they may first seem. A complex number consists of a real part and an imaginary part. Imaginary numbers arise when we take the square root of a negative real number. These solutions or roots are plotted on what is known as the complex plane.

The complex plane is a two-dimensional plane similar to the Cartesian coordinate system used for graphing real numbers. On this plane, the horizontal axis represents the real part of the complex numbers, and the vertical axis represents the imaginary part. For a complex number \( a + bi \), \( a \) marks its position on the horizontal axis, and \( b \) on the vertical axis. If a quadratic equation has complex solutions, they will appear along the imaginary axis, with opposite imaginary values if they are conjugates like \( i\sqrt{3} \) and \(-i\sqrt{3} \).

This visualization helps us better understand and interpret the nature of these complex solutions beyond mere algebraic expressions.

Square Root of Negative Numbers

The concept of taking the square root of a negative number was a mystery until the introduction of imaginary numbers. The square root of \(-1\) is defined as \( i \), an imaginary unit that allows us to extend the concept of square roots to negative numbers. When we encounter \( \sqrt{-b} \) in our calculations, where \( b > 0 \), this becomes \( i\sqrt{b} \).

In the context of quadratic equations, this discovery is crucial when solving equations with no real roots. For instance, when we isolated \( x^2 = -3 \), finding \( x \) involved calculating \( \sqrt{-3} \), leading to the solutions \( x = \pm i\sqrt{3} \).

Understanding how imaginary numbers intersect with real numbers on the complex plane not only expands our comprehension of number systems but also enhances our problem-solving abilities in various branches of mathematics and applied sciences.