Question 255656
{{{x^2 -4 = x}}}
<pre><font size = 4 color = "indigo"><b>
Get 0 on the right by adding -x to both sides:

{{{x^2-4-x=0}}}

Let y=0

Then we can write the above as this system of equations:

{{{system(y=x^2-4-x,y=0)}}}

Now we draw the graph of the first equation by getting
some points:

 x | y
------
-3 | 8
-2 | 2
-1 |-2
 0 |-4
 1 |-4
 2 |-2 
 3 | 2
 4 | 8

{{{drawing(257.1,400,-4,5,-5,9,
graph(257.2,400,-4,5,-5,9),
line(-3 +.1, 8, -3 -.1,8), line(-3, 8 +.1, -3, 8 -.1),
line(-3 +.1, 8 +.1, -3 -.1, 8-.1), line(-3 +.1, 8 -.1, -3 -.1, 8+.1) 

line(-2 +.1, 2, -2 -.1,2), line(-2, 2 +.1, -2, 2 -.1),
line(-2 +.1, 2 +.1, -2 -.1, 2-.1), line(-2 +.1, 2 -.1, -2 -.1, 2+.1)

line(-1 +.1,-2, -1 -.1,-2), line(-1, -2 +.1, -1, -2 -.1),
line(-1 +.1, -2 +.1, -1 -.1, -2-.1), line(-1 +.1, -2 -.1, -1 -.1, -2+.1)

line(0 +.1, -4, 0 -.1,-4), line(0, -4 +.1, 0, -4 -.1),
line(0 +.1, -4 +.1, 0 -.1, -4-.1), line(0 +.1,-4 -.1, 0 -.1, -4+.1)

line(1 +.1, -4, 1 -.1,-4), line(1, -4 +.1, 1, -4 -.1),
line(1 +.1, -4 +.1, 1 -.1, -4-.1), line(1 +.1, -4 -.1, 1 -.1, -4+.1)

line(2 +.1, -2, 2 -.1,-2), line(2, -2 +.1, 2, -2 -.1),
line(2 +.1, -2 +.1, 2 -.1, -2-.1), line(2 +.1, -2 -.1, 2 -.1, -2+.1)

line(3 +.1, 2, 3 -.1,2), line(3, 2 +.1, 3, 2 -.1),
line(3 +.1, 2 +.1, 3 -.1, 2-.1), line(3 +.1, 2 -.1, 3 -.1, 2+.1)

line(4 +.1, 8, 4 -.1,8), line(4, 8 +.1, 4, 8 -.1),
line(4 +.1, 8 +.1, 4 -.1, 8-.1), line(4 +.1, 8 -.1, 4 -.1, 8+.1)



)}}}

Draw a smooth curve through those points

{{{drawing(257.1,400,-4,5,-5,9,
graph(257.1,400,-4,5,-5,9, x^2-x-4),
line(-3 +.1, 8, -3 -.1,8), line(-3, 8 +.1, -3, 8 -.1),
line(-3 +.1, 8 +.1, -3 -.1, 8-.1), line(-3 +.1, 8 -.1, -3 -.1, 8+.1) 

line(-2 +.1, 2, -2 -.1,2), line(-2, 2 +.1, -2, 2 -.1),
line(-2 +.1, 2 +.1, -2 -.1, 2-.1), line(-2 +.1, 2 -.1, -2 -.1, 2+.1)

line(-1 +.1,-2, -1 -.1,-2), line(-1, -2 +.1, -1, -2 -.1),
line(-1 +.1, -2 +.1, -1 -.1, -2-.1), line(-1 +.1, -2 -.1, -1 -.1, -2+.1)

line(0 +.1, -4, 0 -.1,-4), line(0, -4 +.1, 0, -4 -.1),
line(0 +.1, -4 +.1, 0 -.1, -4-.1), line(0 +.1,-4 -.1, 0 -.1, -4+.1)

line(1 +.1, -4, 1 -.1,-4), line(1, -4 +.1, 1, -4 -.1),
line(1 +.1, -4 +.1, 1 -.1, -4-.1), line(1 +.1, -4 -.1, 1 -.1, -4+.1)

line(2 +.1, -2, 2 -.1,-2), line(2, -2 +.1, 2, -2 -.1),
line(2 +.1, -2 +.1, 2 -.1, -2-.1), line(2 +.1, -2 -.1, 2 -.1, -2+.1)

line(3 +.1, 2, 3 -.1,2), line(3, 2 +.1, 3, 2 -.1),
line(3 +.1, 2 +.1, 3 -.1, 2-.1), line(3 +.1, 2 -.1, 3 -.1, 2+.1)

line(4 +.1, 8, 4 -.1,8), line(4, 8 +.1, 4, 8 -.1),
line(4 +.1, 8 +.1, 4 -.1, 8-.1), line(4 +.1, 8 -.1, 4 -.1, 8+.1)



)}}}
   
Now the graph of the other equation in the system, y=0,
is simply the x-axis, so we only need to know the points where
the graph intersects the x-axis.=, and they will be the
solutions to the system

{{{system(y=x^2-4-x,y=0)}}}

Notice that they cross at the short green lines below:

{{{drawing(257.1,400,-4,5,-5,9,
graph(257.1,400,-4,5,-5,9, x^2-x-4),
line(-3 +.1, 8, -3 -.1,8), line(-3, 8 +.1, -3, 8 -.1),
line(-3 +.1, 8 +.1, -3 -.1, 8-.1), line(-3 +.1, 8 -.1, -3 -.1, 8+.1) 

line(-2 +.1, 2, -2 -.1,2), line(-2, 2 +.1, -2, 2 -.1),
line(-2 +.1, 2 +.1, -2 -.1, 2-.1), line(-2 +.1, 2 -.1, -2 -.1, 2+.1)

line(-1 +.1,-2, -1 -.1,-2), line(-1, -2 +.1, -1, -2 -.1),
line(-1 +.1, -2 +.1, -1 -.1, -2-.1), line(-1 +.1, -2 -.1, -1 -.1, -2+.1)

line(0 +.1, -4, 0 -.1,-4), line(0, -4 +.1, 0, -4 -.1),
line(0 +.1, -4 +.1, 0 -.1, -4-.1), line(0 +.1,-4 -.1, 0 -.1, -4+.1)

line(1 +.1, -4, 1 -.1,-4), line(1, -4 +.1, 1, -4 -.1),
line(1 +.1, -4 +.1, 1 -.1, -4-.1), line(1 +.1, -4 -.1, 1 -.1, -4+.1)

line(2 +.1, -2, 2 -.1,-2), line(2, -2 +.1, 2, -2 -.1),
line(2 +.1, -2 +.1, 2 -.1, -2-.1), line(2 +.1, -2 -.1, 2 -.1, -2+.1)

line(3 +.1, 2, 3 -.1,2), line(3, 2 +.1, 3, 2 -.1),
line(3 +.1, 2 +.1, 3 -.1, 2-.1), line(3 +.1, 2 -.1, 3 -.1, 2+.1)

line(4 +.1, 8, 4 -.1,8), line(4, 8 +.1, 4, 8 -.1),
line(4 +.1, 8 +.1, 4 -.1, 8-.1), line(4 +.1, 8 -.1, 4 -.1, 8+.1),

green(line(-1.561553,1,-1.561553,-1)),

green(line(2.5615528,1,2.5615528,-1))

)}}}

The x-intercept on the left is a little more than half way between
-1 and -2, so "guess"timate it to be -1.6.

The x-intercept on the right is a little more than half way between
2 and 3, so "guess"timate it to be 2.6.

So the solutions are about x = -1.6 and x = 2.6.

Edwin</pre>