Question 943223
I can show you the way I do it
{{{ -4x^2 + 3x + 7 = 0 }}}
Subtract {{{ 7 }}} from both sides
{{{ -4x^2 + 3x = -7 }}}
Divide both sides by {{{ -4 }}}
{{{ x^2 -( 3/4 )*x = (-7)/(-4) }}}
{{{ x^2 -( 3/4 )*x = 7/4 }}}
Take {{{ 1/2 }}} of the coefficient of the
{{{ x }}} term, then square it, then add it
to both sides
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{{{ x^2 -(3/4)*x + (-3/8)^2 = 7/4 + (-3/8)^2 }}}
{{{ x^2 -(3/4)*x + 9/64 = 7/4 + 9/64 }}}
{{{ x^2 -(3/4)*x + 9/64 = 112/64 + 9/64 }}}
{{{ x^2 -(3/4)*x + 9/64 = 121/64 }}}
Believe it or not, both sides are now perfect squares!
{{{ ( x - 3/8 )^2 = ( 11/8 )^2 }}}
Take the square root of both sides
{{{ x - 3/8 = 11/8 }}}
{{{ x = 14/8 }}}
And, using the negative square root of the right side,
{{{ x - 3/8 = -11/8 }}}
{{{ x = -1 }}}
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So, the factoring would be:
{{{ ( x - 14/8 )*( x + 1 ) = 0 }}}
{{{ x^2 - (14/8)*x + x - 14/8 = 0 }}}
{{{ x^2 - ( 6/8 )*x - 14/8 = 0 }}}
Multiply both sides by {{{ -8 }}}
{{{ -8x^2 + 6x + 14 = 0 }}}
Divide both sides by {{{ 2 }}}
{{{ -4x^2 + 3x + 7 = 0 }}}
whew! This is a difficult one, but the
method is always the same