Problem 19 - Precalculus : Building Concepts and Connections

Question

Use the definition of i to solve the equation. $$3 x^{2}=-30$$

Step-by-Step Solution

Answer

The solutions are \(x = \pm \sqrt{10}i\).

Step 1: Rewrite the equation

First, the equation needs to be rewritten such that \(x^{2}\) is on one side and the constants on the other side. So, rewrite the equation as \(x^{2} = -10\)

Step 2: Take square root on both sides

Take the square root on both sides of the equation leads to \(x = \sqrt{-10}\). But, the square root of a negative number is not possible in the real number field and that's where the \(i\) comes into play.

Step 3: Use the definition of i

According to the definition of \(i\), \(i^2 = -1\). Hence, \(x = \sqrt{-10}\) can be rewritten as \(x = \sqrt{10} \cdot i\).

Step 4: Simplify

The square root of 10 cannot be simplified further, hence the final answers are \(x = \pm \sqrt{10}i\). Here, \(\pm\) indicates that \(x\) could be either \(+\sqrt{10}i\) or \(-\sqrt{10}i\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is foundational in understanding complex numbers. The imaginary unit is denoted by the symbol \(i\) and is defined by the property \(i^2 = -1\). This definition allows mathematicians to extend the real number system to include solutions to equations that don't have solutions in the real numbers alone.

Consider the equation \(x^2 + 1 = 0\). In the realm of real numbers, this equation has no solution because no square of a real number can be negative. However, using the definition of \(i\), we realize that \(x^2 = -1\) can be satisfied by \(x = i\) or \(x = -i\).

The introduction of the imaginary unit \(i\) enables us to work with numbers and equations that involve the square roots of negative numbers. It plays a key role in the field of complex numbers, which includes expressions like \(a + bi\), where \(a\) and \(b\) are real numbers.

Square Root of Negative Numbers

Square roots of negative numbers can be tricky when first encountered in mathematics. In the set of real numbers, the square root of a negative number does not exist. This is because squaring a real number, whether positive or negative, always results in a positive number.

To handle square roots of negative numbers, we extend the number system by introducing the imaginary unit \(i\), where \(i^2 = -1\). Thus, the square root of a negative number can be expressed using \(i\). For instance:

• \(\sqrt{-4} = \sqrt{4} \cdot i = 2i\)
• \(\sqrt{-9} = \sqrt{9} \cdot i = 3i\)

This approach helps in simplifying expressions and solving algebraic equations that involve negative square roots.

When solving equations, it is crucial to separate the negative sign first, before dealing with the square root. For example, \(\sqrt{-10}\) can be rewritten as \(\sqrt{10} \cdot i\), indicating that the solution involves an imaginary component.

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). Solving these equations can be straightforward when they have real solutions, but sometimes they require the use of complex numbers, especially when the discriminant (\(b^2 - 4ac\)) is negative.

To solve a quadratic equation with a negative discriminant, the use of the imaginary unit \(i\) becomes necessary. Let's explore this via a straightforward example:

• Suppose we have an equation like \(x^2 = -10\), as in our original exercise.
• First, isolate \(x^2\) on one side: \(x^2 = -10\).
• Then, take the square root of both sides: \(x = \pm \sqrt{-10}\).
• Since \(\sqrt{-10}\) involves a negative number under the square root, express it using \(i\) as \(x = \pm \sqrt{10}i\).

Quadratic equations may not always yield real solutions. When they involve square roots of negative numbers, the solutions are expressed in terms of complex numbers, demonstrating the indispensable role of \(i\) in extending the number line to encompass all possible solutions.