Question

What is the solution of the equation when solved over the complex numbers? X2+24=0

Answer 1

The solution to the equation x² + 24 = 0 over the complex numbers is x = ±2√(6)i. After isolating x² on one side, we take the square root of both sides and introduce the imaginary unit i because the number under the square root is negative.

Step-by-step explanation:

To solve the equation x² + 24 = 0 over the complex numbers, we need to find values of x that satisfy the equation. This is a quadratic equation with 'a' as 1 (the coefficient of x²), 'b' as 0 (since there is no x term), and 'c' as 24.

We will first isolate x² by subtracting 24 from both sides of the equation:

x² = -24

Next, to find x, we take the square root of both sides. Remember that when we take the square root of a negative number, we get an imaginary number. The square root of -24 can be expressed as:

x = ±√(-24)

This simplifies to:

x = ±√(24) × ±i

Since √(24) is √(4 × 6), which simplifies to 2√(6), the final solution in terms of complex numbers is:

x = ±2√(6)i

Answer 2

You can rearrange the equation as follows:

`x^2 = -24`

Using real numbers, it would be impossible for a square to be negative, but using imaginary numbers it is possible, since
`i^2 = -1`

So, since
`√(24)^2 = 24`, we know that

`(i√(24))^2 = i^2\cdot √(24)^2 = (-1)\cdot 24 = -24`

So, the solutions to the equations are

`x^2 = -24 \iff x = \pm√(-24) = \pm i √(24)`