Question 570907
{{{ f(x) = 64x^2 + b*x + 16 }}}
The quadratic formula to find roots is:
{{{x = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 64 }}}
{{{ b = b }}}
{{{ c = 16 }}}
To make {{{ f(x) }}} a perfect square, {{{ b^2 - 4*a*c }}}
must be zero
 {{{ b^2 - 4*a*c  = 0 }}} 
{{{ b^2 - 4*64*16 = 0 }}}
{{{ b^2 - 4096 = 0 }}}
{{{ b^2 = 4096 }}}
{{{ b = 64 }}}
{{{ b = -64 }}}
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If {{{ b = 64 }}}
{{{ f(x) = 64x^2 + b*x + 16 }}}
{{{ f(x) = 64x^2 + 64x + 16 }}}
{{{ f(x) = 4x^2 + 4x + 1 }}}
{{{ f(x) = ( 2x + 1 )^2 }}}
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If {{{ b = -64 }}}
{{{ f(x) = 64x^2 - 64x + 16 }}}
{{{ f(x) = 4x^2 - 4x + 1 }}}
{{{ f(x) = ( 2x - 1 )^2 }}}
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{{{b}}} can be replaced with either {{{ 64 }}} or {{{ -64 }}}
Here are plots of the 2 alternatives:
{{{ graph( 400, 400, -4, 4, -4, 8, 4x^2 + 4x + 1, 4x^2 - 4x + 1 ) }}}