-2x2-x-1=0
our answer...

→   x  = - 1 ±  i√7     ;

                  4    

Or:  Write as:

 →   x  = -1  +  i√7     ;   -1  −   i√7     ;

                   4                       4  

Step-by-step explanation:

Given:

 -2x² − x − 1 = 0 ;

Solve for " x " .

First, let us multiply BOTH sides of the equation by "-1" ;

     to get rid of that "negative sign" in the coefficient:  " -2" ;

    →  as follows:

  -1 * {-2x² − x − 1 } = 0 *{-1} ;

Rewrite the value: "x" on the "left-hand side" of the question; as:  "1x" ;

since "any value" ; multipled by "1" ; results in that same individual value:

    -1 * {-2x² − 1x − 1 } = 0 *{-1} ;  

Note the "distributive property" of multiplication:

 a(b + c) = ab + ac ;

Likewise, from the "left-hand side" of the equation:

   -1 * {-2x² − 1x − 1 } =  {-1 * -2x²}  +  {-1 * -1x}  +  {-1* -1 } ;    

                                =      2x²       +  1x           + 1 ;

Rewrite the "left-hand side" of the equation:

   2x² + 1x + 1 ;

Now, the "right-hand side of the equation:

            " { 0 * -1 } = 0 " .

Now, rewrite the entire equation:

    2x² + 1x + 1 = 0 ;

Note:  This equation is written in "quadratic format" ;

that is:   " ax² + bx + c " ;   [a ≠ 0] ;

         →  in which:

                   a = 2 ;  b = 1 ; c = 1 ;

 →  which means we can solve for the value(s) for "x" ; by using the

"quadratic equation formula" ;

   →  that is:

   →    x  = -b ± √(b² − 4ac)       ;

                           2a                      

 So;  to solve for the values for "x" ;  we substitute our known values for:

    "a" ; "b" ; and "c" ; and solve:

   →   x  =   -1  ±  √(1² − 4*2 *1)       ;

                              4                    

   →   x   =    -1  ±  √(1 − 8)       ;

                            4                              

   →   x  =   -1  ±  √(-7)     ;

                         4        

Note: "√(-7)"  ;  is the square root of a negative number.

Note that we use the lower case  letter, " i "  ;  

        as a symbol to represent the imaginary number:  " √(-1) " .

This is an "imaginary number" ;  since one cannot take the "square root" of a "number lower than zero."  

So, we write:  √-7 ;   as:   " i * 7 " ;  or, just:  " i √7 " ;  since:

  "i  = √-1" ;  and since:  " √-7"  would "factor out" to:  "√7  * √-1 " .

Note:  In the mainstream U.S.A. , the concept of " i " as the  "imaginary number" — " √-1 "  ;  is usually introduced in the second year college-prep algebra class (after "first year high school algebra" in 9th grade; and after "geometry" in 10th grade).

So;  we have:

→   x  = -1  ± √(-7)     ;

                   4        

We can rewrite this as:

→   x  = -1  ±  i√7    ;

                   4    

  We can leave our answer written like that.

We have two (2) solutions—so we can also write out both solutions for the value of x:

 →   x  = -1  +  i√7    ;   -1  +  i√7     ;

                   4                        4

Hope this explanation is helpful to you!

  Best of luck— and best wishes!

sup

step-by-step explanation: