Question 703926
<pre>
x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0

 

There are two ways to solve this:

Method 1:  By u-substitution:

let u = &#8730;<span style="text-decoration: overline">x</span>
then u² = x

Substitute u² for x and u for &#8730;<span style="text-decoration: overline">x</span>

u² - 2u - 80 = 0

That factors as:

(u - 10)(u + 8) = 0

u = 10, u = -8

Substitute &#8730;<span style="text-decoration: overline">x</span> for u

   &#8730;<span style="text-decoration: overline">x</span> = 10           &#8730;<span style="text-decoration: overline">x</span> = -8
(&#8730;<span style="text-decoration: overline">x</span>)² = 10²       (&#8730;<span style="text-decoration: overline">x</span>)² = (-8)²
    x = 100           x = 64

Check for extraneous solution(s):

     x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0
 100 - 2&#8730;<span style="text-decoration: overline">100</span> - 80 = 0
 100 - 2(10) - 80 = 0
    100 - 20 - 80 = 0
                0 = 0

That's true, so x=10 is a solution.

     x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0
   64 - 2&#8730;<span style="text-decoration: overline">64</span> - 80 = 0
   64 - 2(8) - 80 = 0
     64 - 16 - 80 = 0
              -32 = 0

That's false, so the only solution is x=10



------------------------------
Method 2:  By isolating the radical term and squaring
both sides:

    x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0
          x - 80 = 2&#8730;<span style="text-decoration: overline">x</span>
       (x - 80)² = 4(&#8730;<span style="text-decoration: overline">x</span>)²
x² - 160x + 6400 = 4x 
x² - 164x + 6400 = 0
   (x-100)(x-64) = 0

x = 100, x = 64

Check for extraneous solution(s):

     x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0
 100 - 2&#8730;<span style="text-decoration: overline">100</span> - 80 = 0
 100 - 2(10) - 80 = 0
    100 - 20 - 80 = 0
                0 = 0

That's true, so x=10 is a solution.

     x - 2&#8730;<span style="text-decoration: overline">x</span> - 80 = 0
   64 - 2&#8730;<span style="text-decoration: overline">64</span> - 80 = 0
   64 - 2(8) - 80 = 0
     64 - 16 - 80 = 0
              -32 = 0

That's false, so the only solution is x=10

Edwin</pre>