Question

What is the solution of the equation x2 + 4 = 0?

Answer 1

Answer:

Two solutions were found :

 x=  0.0000 - 2.0000 i

 x=  0.0000 + 2.0000 i

Step-by-step explanation:

Find roots (zeroes) of :       F(x) = x2+4

Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  4.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,2 ,4

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        5.00    

     -2       1        -2.00        8.00    

     -4       1        -4.00        20.00    

     1       1        1.00        5.00    

     2       1        2.00        8.00    

     4       1        4.00        20.00    

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

 x2 + 4  = 0

Solving a Single Variable Equation :

2.1      Solve  :    x2+4 = 0

Subtract  4  from both sides of the equation :

                     x2 = -4

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  

                     x  =  ± √ -4  

In Math,  i  is called the imaginary unit. It satisfies   i2  =-1. Both   i   and   -i   are the square roots of   -1

Accordingly,  √ -4  =

                   √ -1• 4   =

                   √ -1 •√  4   =

                   i •  √ 4

Can  √ 4 be simplified ?

Yes!   The prime factorization of  4   is

  2•2

To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 4   =  √ 2•2   =

               ±  2 • √ 1   =

               ±  2

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                     x=  0.0000 + 2.0000 i

                     x=  0.0000 - 2.0000 i

Two solutions were found :

 x=  0.0000 - 2.0000 i

 x=  0.0000 + 2.0000 i

Processing ends successfully

Answer 2

Answer:

x = ± 2i

Step-by-step explanation:

x² + 4 = 0 has no real solutions but has 2 complex solutions

Note that $\sqrt{-1}$ = i

Given

x² + 4 = 0 ( subtract 4 from both sides )

x² = - 4 ( take the square root of both sides )

x = ± $\sqrt{-4}$

  = ± $\sqrt{4(-1)}$

  = ± $\sqrt{4}$ × $\sqrt{-1}$

  = ± 2i

Thus the solutions are x = - 2i or x = 2i