Problem 54 - Algebra and Trigonometry

Question

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 9 x^{2}+4=0 $$

Step-by-Step Solution

Answer

The solutions are \(0 + \frac{2}{3}i\) and \(0 - \frac{2}{3}i\).

Step 1: Set up the Equation

The given equation is \(9x^2 + 4 = 0\). To solve for \(x\), we first need to isolate the \(x\) term.

Step 2: Move Constant to the Other Side

Subtract 4 from both sides of the equation, resulting in: \(9x^2 = -4\). This makes the coefficient of \(x^2\) positive on one side and the constant negative on the other.

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

Divide both sides by 9 to solve for \(x^2\), giving \(x^2 = -\frac{4}{9}\). This isolates \(x^2\) on one side of the equation.

Step 4: Take the Square Root of Both Sides

To find \(x\), take the square root of both sides of the equation: \(x = \pm \sqrt{-\frac{4}{9}}\). Recall that the square root of a negative number involves \(i\), the imaginary unit.

Step 5: Simplify the Expression

The square root of \(-1\) is \(i\), so \(x = \pm \sqrt{\frac{4}{9}} \cdot i = \pm \frac{2}{3}i\). This results from simplifying the square root of the fraction.

Step 6: Express Solutions in Form \(a + bi\)

The solutions of the equation are \(x = 0 + \frac{2}{3}i\) and \(x = 0 - \frac{2}{3}i\). This shows both the positive and negative imaginary solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation that takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This specific form tells us that the equation involves a variable raised to the second power, making it a 'quadratic'. These equations can be solved either by:

• Factoring the equation if possible.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square to transform it into a simpler form.
• Graphically, to find points where the graph of the equation crosses the x-axis.

The equation in our example, \(9x^2 + 4 = 0\), is already simplified, but it does not have a linear term (where it doesn't include \(x\) alone), which makes it a simpler problem to tackle directly. Usually, the goal is to solve for \(x\) by rearranging terms and applying necessary algebraic operations.

Imaginary Unit

The imaginary unit, usually denoted by \(i\), is a mathematical concept used to extend the real number system. It is defined by the property \(i^2 = -1\). This property allows mathematicians to work with and find solutions for the square roots of negative numbers, which are not possible within the scope of real numbers. Here are some key points about the imaginary unit:

• The imaginary unit is not a real number, but it is used to express complex numbers.
• Complex numbers are of the form \(a + bi\), where \(a\) and \(b\) are real numbers.
• The number \(a\) is called the real part, and \(b\) is called the imaginary part.
• An example is \(0 + \frac{2}{3}i\), where \(0\) is the real part and \(\frac{2}{3}i\) is the imaginary part.

In the context of the given exercise, the imaginary unit becomes crucial when solving \(x^2 = -\frac{4}{9}\) because it involves the square root of a negative number. By recognizing \(-\frac{4}{9}\) as \(\left(-1\right) \cdot \frac{4}{9}\), we can use \(i\) to express the square root as \(x = \pm \frac{2}{3}i\).

Solving Equations

When we talk about solving equations, we mean finding the values for the variable that make the equation true. In our example, we aim to find \(x\) such that the equation \(9x^2 + 4 = 0\) holds. This involves several steps, including:

• Isolating terms to make the expression simpler. For quadratic equations, this often means isolating the squared term, as shown when we moved 4 to the other side: \(9x^2 = -4\).
• Solving for the isolated term. Dividing both sides by 9 isolates \(x^2\), resulting in \(x^2 = -\frac{4}{9}\).
• Taking square roots to solve for \(x\) (consider both positive and negative roots). Since \(x^2 = -\frac{4}{9}\), we find that \(x = \pm \frac{2}{3}i\).
• Using algebraic rules and properties, such as recognizing that the square root of a negative number involves \(i\).

The final solutions \(x = 0 + \frac{2}{3}i\) and \(x = 0 - \frac{2}{3}i\) represent the form \(a + bi\), typical of complex numbers, reflecting both positive and negative imaginary components.