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The Quadratic Formula

Here's one that doesn't factor:

Solve  x^2 + x - 1 = 0

a = 1 , b = 1 , c = -1
 

x = -b +/- sqrt( b^2 - 4ac ) / 2a = -1 +/- sqrt( ( 1 )^2 - 4 ( 1 ) ( -1 ) ) /( 2 ( 1) ) = -1 +/- sqrt( 1 + 4 ) / 2 = -1 +/- sqrt( 5 ) / 2

So x = -1 - sqrt( 5 ) / 2 or x = -1 + sqrt( 5 ) / 2

Yep -- these messy little buggers are the answers:

{ -1 - sqrt( 5 ) / 2 , -1 + sqrt( 5 ) / 2 }

If you were in a science or business class, you'd need to pop these into your calculator to get decimal approximations for these.  Since you may need to do this someday, we better practice this. Since all calculators are different, it won't do much good for me to tell you how to enter this in... So, you're on your own.  Here are the answers:

exact: { -1 - sqrt( 5 ) / 2 , -1 + sqrt( 5 ) / 2 }

decimal approximation: { -1.618 , .618 }

Find out which of these your teacher wants.