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

Question

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

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Sadie 2 weeks 2022-01-08T18:23:00+00:00 1 Answer 0 views 0

Answers ( )

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    2022-01-08T18:24:29+00:00

    Answer:

    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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