Question

Solve each equation in the complex number system. $$ x^{2}+25=0 $$

Short answer

x = ±5i

Step-by-step solution

Move the constant term

To solve the equation, first isolate the term with the variable on one side of the equation. So, subtract 25 from both sides: x^{2} + 25 - 25 = 0 - 25 so x^{2} = -25

Take the square root

Take the square root of both sides of the equation to solve for x. Remember that squaring and taking the square root are inverse operations and you should consider both the positive and negative roots: x = sqrt {25} = ± sqrt {25} i = ± 5i =25 x = ± 5i

Key concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation in which the highest power of the variable is 2. The general format is ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. To solve quadratic equations, you can use a variety of methods, such as:

• Factoring
• Completing the square
• Using the quadratic formula
• Graphing

When the solutions are not real numbers, they usually involve complex numbers. Quadratic equations often have two solutions, which can be real or complex. In our given exercise, we have: x^2 + 25 = 0. First, we move the 25 to the other side of the equation: x^2 = -25.Next, we take the square root of both sides, which leads us to complex numbers.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. A basic complex number looks like this: a + bi, where a is the real part and bi is the imaginary part. The imaginary unit 'i' is defined as the square root of -1, which means that i^2 = -1. Complex numbers enable us to solve equations that don't have real solutions. In our exercise, we find x by taking the square root of a negative number, -25. This involves using the imaginary unit 'i': sqrt(-25) = ± 5i. Here ± indicates two solutions, +5i and -5i. Complex numbers are essential in advanced mathematics and engineering, making them important to understand.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 * 5 = 25. When solving quadratic equations, taking the square root is a crucial step. But what happens if the number is negative? We use the imaginary unit 'i'. For instance: sqrt(-25) involves finding the square root of 25 and multiplying by 'i', giving us ±5i. When solving quadratics, always remember that taking the square root introduces both a positive and a negative solution. In summary, understanding square roots, including those of negative numbers, is vital for solving quadratic equations in the complex number system.