Problem 66 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$x^{2}+4.0=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 2i\) and \(x = -2i\).

Step 1: Rearrange the Equation

First we rearrange the given equation \(x^{2}+ 4.0 = 0\) to make it in the form something equals zero. This can be easily done by subtracting 4.0 from both sides of the given equation which gives the new equation as \(x^{2} = -4.0\)

Step 2: Obtain the square root of the equation

To solve the value of \(x\), we need to find the square root of both sides. By doing so, we get \(x = \sqrt{-4.0}\)

Step 3: Write the Solution as a Complex Number

The square root of a negative number is not defined in the real numbers, but it is defined in the complex numbers. The square root of -1 is 'i' in the complex numbers, therefore, we can rewrite the solution as \(x = \sqrt{4.0}\times i = 2.0i\). Since the squaring operation comes with a positive and a negative result, the final solution for the equation is \(x = 2i, x = -2i\)

Key Concepts

Understanding Complex Numbers

When solving equations like \(x^2 + 4 = 0\), you might encounter something called complex numbers. Complex numbers extend the concept of the real number system. A complex number is in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). This definition allows for the square roots of negative numbers to be calculated, which cannot be done in the real number system. The result is a whole new way to solve equations that would otherwise have no real solutions.

Calculating Square Roots of Negative Numbers

Square roots are typically used to find a number that, when multiplied by itself, gives the original number. The square root of a positive number results in a real number. However, when dealing with negative numbers, the notion of square roots shifts to include complex numbers. When you encounter a square root of a negative number like \(\sqrt{-4}\), it can be challenging because real numbers multiplied by themselves can never be negative. This is where the imaginary unit, \(i\), comes into play. By setting \(\sqrt{-1} = i\), we can compute \(\sqrt{-4}\) as follows: since \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2 \times i = 2i\). This is why the solutions to \(x^2 = -4\) include complex numbers, specifically \(2i\) and \(-2i\).

The Role of Imaginary Numbers

Imaginary numbers are essential when it comes to solving quadratic equations involving negative square roots. Imaginary numbers have a crucial role in mathematics and engineering. They enable the solutions of equations that would otherwise have no solutions in the real number system. For instance, the earlier equation \(x^2 + 4 = 0\) is unsolvable using just real numbers. However, by incorporating the imaginary unit \(i\), we can express the solutions as \(x = 2i\) and \(x = -2i\).

• Imaginary numbers are used in various fields such as signal processing and electrical engineering.
• They help simplify problems and solutions in calculus and differential equations.
• Imaginary numbers are also instrumental in quantum physics and applied mathematics.

Remember, without imaginary numbers, certain mathematical calculations would be impossible, and many areas of science and technology would not be as advanced as they are today.