Question 171507
You were headed down the right path but made one little mis-step.

Your evaluation of the quadratic formula was correct up to a point, so let's review:

{{{x = (-b +- sqrt( b^2-4ac ))/(2a) }}}

{{{x = (-8 +- sqrt( 8^2-4*1*(-168) ))/(2*1) }}}

{{{x = (-8 +- sqrt( 736 ))/2 }}}  So far, so good.

But then you simplified to  {{{x = -4 +- sqrt( 736 ) }}} which is incorrect because you only divided ONE of the numerator terms by the denominator.  That is against the rules:  the other team gets a penalty kick.

What you need to do is find the prime factorization of 736.

736/2 = 368
368/2 = 184
184/2 = 92
92/2 = 46
46/2 = 23
23 is prime

Now you can say {{{736=23 * 2 * 2^4}}}  and {{{sqrt(736)=sqrt(23 * 2 * 2^4)=4*sqrt(46)}}}

Going back to where you were before you went astray:

{{{x = (-8 +- sqrt( 736 ))/2 }}}

{{{x = (-8 +- 4*sqrt( 46 ))/2 }}}

{{{x = -4 +- 2*sqrt( 46 ) }}}

{{{ -4 - 2*sqrt( 46 ) }}} is negative, so exclude this root as extraneous.

Evaluate {{{-4 + 2*sqrt( 46 ) }}} which gets you approximately 9.56, the length of the short leg of the triangle.  Since the longer leg is 8 feet longer, the longer leg must be approximately 17.56 feet.  If you walk the sidewalk, you have to go 17.56 feet plus 9.56 feet or 27.12 feet.  If you walk the diagonal, you only have to walk 20 feet.  The difference is 27.12 - 20 = 7.12.  Rounded to the nearest foot, 7 feet.

Notice that I didn't do the round-off until the very last step.  Whenever you round numbers, you introduce errors, so it is important to keep greater precision (typically 1 or 2 decimal places greater precision than the precision required in the answer) until you have made your final calculation.