Problem 53 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions are \(0 + 3i\) and \(0 - 3i\).

Step 1: Rewrite the equation in standard quadratic form

The given equation is \(x^2 + 9 = 0\). Notice that this equation is already in the standard quadratic form \(ax^2 + bx + c = 0\) with \(a = 1\), \(b = 0\), and \(c = 9\). In this case, the equation is \(x^2 + 0x + 9 = 0\).

Step 2: Isolate the quadratic term

Next, isolate the \(x^2\) term by subtracting 9 from both sides of the equation: \(x^2 = -9\).

Step 3: Solve for \(x\) by taking the square root

Take the square root of both sides of the equation \(x^2 = -9\): \(x = \pm \sqrt{-9}\).

Step 4: Simplify the square root

Since the square root of a negative number involves the imaginary unit \(i\), rewrite \(\sqrt{-9}\) as \(\sqrt{9} \cdot \sqrt{-1}\). Hence, \(\sqrt{-9} = 3i\).

Step 5: Express the solutions in the form \(a + bi\)

The solutions from the previous step are \(x = 3i\) and \(x = -3i\). Expressed in the form \(a + bi\), they are \(0 + 3i\) and \(0 - 3i\).

Key Concepts

Imaginary Numbers

When solving the equation \(x^2 + 9 = 0\), you come across the need to take the square root of a negative number. This is where imaginary numbers come into play. Imaginary numbers arise when you find the square root of a negative number, which is not possible within the realm of real numbers. To tackle this, mathematicians introduced the imaginary unit, denoted by \(i\), which is defined as \(i = \sqrt{-1}\).

By using the imaginary unit, the square root of any negative number can be expressed in terms of \(i\). For example, \(\sqrt{-9}\) can be rewritten as \(\sqrt{9} \times \sqrt{-1} = 3i\). Imaginary numbers are an extension of the real number system and are written in the form of \(bi\), where \(b\) is a real number.

Complex Numbers

Combining real numbers with imaginary numbers results in complex numbers. A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Thus, complex numbers form a set of numbers that include both real numbers and imaginary numbers.

In the context of the equation \(x^2 + 9 = 0\), after solving and finding \(x = 3i\) and \(x = -3i\), you can express these solutions as complex numbers. Specifically, they are \(0 + 3i\) and \(0 - 3i\), where \(a\) (the real part) is 0 and \(b\) (the imaginary part) is \(\pm 3\).

Understanding complex numbers enables you to solve more complex equations that involve both real and imaginary quantities.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula is specifically helpful when equations cannot be easily factored. The quadratic formula states that \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In some cases, the equation might lead to a negative number under the square root, which results in imaginary solutions. However, for the equation \(x^2 + 9 = 0\), it was solved without applying the quadratic formula directly, as it was simpler to directly isolate \(x^2\) and solve for \(x\). Nonetheless, understanding the quadratic formula is essential as it lays the foundation for solving any quadratic equation, including those with complex solutions.