Solve x^2= -75, where x is a real number.
Simplify your answer as much as possible.

x=  0.0000 - 8.6603

x=  0.0000 + 8.6603

Step-by-step explanation:

Step  1  :

Polynomial Roots Calculator :

1.1    Find roots (zeroes) of :       F(x) = x2+75

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  75.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,3 ,5 ,15 ,25 ,75

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        76.00    

     -3       1        -3.00        84.00    

     -5       1        -5.00        100.00    

     -15       1       -15.00        300.00    

     -25       1       -25.00        700.00    

     -75       1       -75.00        5700.00    

     1       1        1.00        76.00    

     3       1        3.00        84.00    

     5       1        5.00        100.00    

     15       1        15.00        300.00    

     25       1        25.00        700.00    

     75       1        75.00        5700.00    

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

 x2 + 75  = 0

Step  2  :

Solving a Single Variable Equation :

2.1      Solve  :    x2+75 = 0

Subtract  75  from both sides of the equation :

                     x2 = -75

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

                     x  =  ± √ -75  

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

Accordingly,  √ -75  =

                   √ -1• 75   =

                   √ -1 •√  75   =

                   i •  √ 75

Can  √ 75 be simplified ?

Yes!   The prime factorization of  75   is

  3•5•5

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).

√ 75   =  √ 3•5•5   =

               ±  5 • √ 3

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

                     x=  0.0000 + 8.6603 i

                     x=  0.0000 - 8.6603 i

Two solutions were found :

 x=  0.0000 - 8.6603 i

 x=  0.0000 + 8.6603 i

Processing ends successfully

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for ex: the reciprocal of 3/4 is 4/3. it is the opposite.

reciprocal definition:

a mathematical expression or function so related to another that their product is one; the quantity obtained by dividing the number one by a given quantity.