Problem 59 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}=-9$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=-9\) are \(x = 3i\) and \(x = -3i\).

Step 1: Understanding the equation

The equation given is \(x^{2}=-9\). In order to solve for \(x\), one must take the square root of both sides.

Step 2: Applying square root

Taking the square root of \(x^{2}\) on the left side gives us \(x\). However, taking the square root of \(-9\) on the right side is not a real number. In this case, we must use the concept of imaginary numbers. Remember that \(\sqrt{-1}\) is represented by \(i\). Hence, \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\).

Step 3: Writing out the solutions

The solutions to the equation are \(\pm 3i\). This means \(x = 3i\) and \(x = -3i\).

Key Concepts

Complex Numbers

The concept of complex numbers extends the idea of numbers beyond the real number line. A complex number is a combination of a real part and an imaginary part. It is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

• The real part \(a\) is any real number.
• The imaginary part \(bi\) involves the square root of a negative number.

The imaginary unit \(i\) is defined as the square root of \(-1\). This definition allows us to handle equations that do not have solutions in the set of real numbers, such as those involving a square root of a negative number.

In our exercise, the number \(-9\) required us to use complex numbers to find a solution. This results in two possible solutions, expressed as \(3i\) and \(-3i\). Complex numbers like these are crucial in many fields of science and engineering, as they provide a way to solve a wide range of problems.

Imaginary Numbers

Imaginary numbers arise when we take the square root of a negative number. Historically, the existence of such numbers was met with skepticism because they did not fit within the traditional real number framework.

When we defined \(i\) as the square root of \(-1\), mathematicians created a consistent way to extend the number system. This allows us to perform algebraic operations involving square roots of negative numbers, paving the way for a broader understanding of mathematics.

In the equation \(x^2 = -9\), we utilized \(i\) to solve it. Specifically, taking the square root of \(-9\) involves breaking it into \(\sqrt{9} \cdot \sqrt{-1}\), which translates to \(3i\). Imaginary numbers transform problems where no real solutions exist, creating complex but feasible solutions.

Radical Expressions

Radical expressions involve roots of numbers, where the square root is one of the most common types. A radical expression like \(\sqrt{a}\) asks what number squared will result in \(a\).

Radical expressions extend to both positive and negative numbers, but with negative numbers, they generate imaginary numbers as seen in the equation \(x^2 = -9\). By expressing \(\sqrt{-9}\) as \(\sqrt{9} \cdot \sqrt{-1}\), we handle the negative sign separately by introducing \(i\).

When simplifying such an expression, especially in quadratic equations, it often leads to a complex number. Hence, solutions to these equations might not always be integers but are neatly expressed as radical expressions, embracing the inclusion of complex elements. This approach gives us a complete set of solutions to otherwise insolvable problems involving negative roots.