how it is possible to solvex^2+1=-8 I have no ideea. x^2=-9 impossible?

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

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

 

pohnpei397's profile pic

pohnpei397 | College Teacher | (Level 3) Distinguished Educator

Posted on

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.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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).

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