x^2 + x = 2

Question

It's algebra. The original was a simultaneous equation - xy=2 and x+1=y. I substituted the y in and got x^2 + x = 2. I then tried to solve but all I got was x= the square root of (2-x)

Answer 1

To "solve" means to find the values of the variables that make the equations true.

 

Simultaneous equations are true at the same time (that is, simultaneously).

 

With two independent equations, we may solve for two variables.  The methods are (1) substitution and (2) elimination.

 

You used substitution:

    xy = 2

    x(x+1) = 2             [substitute for y from second equation]

    x² + x = 2

    x² + x - 2 = 0             [subtract 2 from both sides]

   (x+2)(x-1) = 0      [either factor or use quadratic formula; verify using F-O-I-L)

The Multiplicative Property of zero (that is, If you multiply any number by 0, you get 0) means that:

     either (x+2)=0  or (x-1)=0   or both.

  That gives:

       x=-2

       x=1

 

Now, solve for the corresponding value of y (use either original equation since both must be true):

     when x=-2, then y=-1

     when x=1, then y= 0

 

When plotted, these two functions cross at two common ("solution") points:    (-2,-1) and (1,0)

Answer 2

x² + x = 2

x² + x - 2 = 0

(x - 1)(x + 2) = 0

 

{-2, 1}

 

Now can you determine y?