x^2 +1 = -8

==> x^2 = -9

It is impossible if the answer is in the real numbers.

However, there is a set of complex numbers that the number sqrt(-1) has a values which is "i"

==> sqrt-1 = i

Now we will rewrite -9 = -1*9

==> x^2 = 9*-1

Now we will take the square root for both side.

==> x = +- sqrt9*sqrt-1

==> x = +- 3*i

Then we have two complex ( not real) solutions.

**==> x1= 3i and x2= -3i**

You have to use imaginary numbers here.

Remember that i is the square root of -1. Since - 9 is -1*9, then the square root of -9 is the square root of -1 times the square root of 9.

This means that the square root of -9 is i*3.

3i is the square root of -9.

But is impossible to solve as square difference.

It is possible to solve as difference of squares, if we are working in the complex numbers field:

x^2 - (-9) = 0

We'll solve according to the previous rule that -1 = i^2.

We can re-write the equation:

x^2 - (-1)(9) = 0

x^2 - 3^2*i^2 = 0

The difference of squares above will be transformed into the product:

(x - 3i)(x + 3i) = 0

We'll set each factor as zero:

x - 3i = 0

**x = 3i**

x + 3i = 0

**x = -3i**

It is possible to solve as difference of squares, if we are working in the complex numbers field:

x^2 - (-9) = 0

We'll solve according to the previous rule that -1 = i^2.

We can re-write the equation:

x^2 - (-1)(9) = 0

x^2 - 3^2*i^2 = 0

The difference of squares above will be transformed into the product:

(x - 3i)(x + 3i) = 0

We'll set each factor as zero:

x - 3i = 0

**x = 3i**

x + 3i = 0

**x = -3i**

No, it is not impossible. We'll use complex numbers to determine the values of x.

We'll continue from the step you've already reached:

x^2 = -9

The next step is to calculate square root both sides:

sqrt x^2 = sqrt(-9)

x1 = +sqrt (-9)

x2 = -sqrt (-9)

We'll solve the problem recalling that sqrt -1 = i.

x1 = sqrt(-1)*sqrt 9

x1 = 3i

x2 = - sqrt(-1)*sqrt 9

x2 = -3i

We can stop the computing process in the moment we find the value of x1 = 3i. We'll complete automatically with the second root, x2 = -3i (the conjugate of x1 = 3i).