x= Â 0.0000 - 8.6603
x= Â 0.0000 + 8.6603
Step-by-step explanation:
Step  1  :
Polynomial Roots Calculator :
1.1 Â Â Find roots (zeroes) of : Â Â Â F(x) = x2+75
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which  F(x)=0 Â
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 : Â 1
of the Trailing Constant : Â 1 ,3 ,5 ,15 ,25 ,75
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     76.00  Â
   -3    1     -3.00     84.00  Â
   -5    1     -5.00     100.00  Â
   -15    1    -15.00     300.00  Â
   -25    1    -25.00     700.00  Â
   -75    1    -75.00     5700.00  Â
   1    1     1.00     76.00  Â
   3    1     3.00     84.00  Â
   5    1     5.00     100.00  Â
   15    1     15.00     300.00  Â
   25    1     25.00     700.00  Â
   75    1     75.00     5700.00  Â
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: Â
           x  =  ± √ -75 Â
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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