Question 1085127
{{{ 6x^2 + 6x + k = 0 }}}
Use quadratic formula
{{{ x = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 6 }}}
{{{ b = 6 }}}
{{{ c= k }}}
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{{{ x = ( -6 +- sqrt( 6^2 - 4*6*k )) / (2*6) }}}
{{{ x = ( -6 +- sqrt( 36 - 24k )) / 12 }}}
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(a)
2 equal real roots
{{{ 36 - 24k }}} must be zero. 
{{{ 36 - 24k = 0 }}}
{{{ 24k = 36 }}}
{{{ k = 3/2 }}}
The 2 equal real roots are:
{{{ x = -1/2 }}}
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(b)
2 distinct real roots
{{{ 36 - 24k }}} must be positive
I'll say
{{{ 36 - 24k = 36 }}}
{{{ k = 0 }}}
The roots are:
{{{ x = 0 }}}
{{{ x = -1 }}}
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No real roots
{{{ 36 - 24k }}} must be negative
I'll say {{{ 36 - 24k = -36 }}}
{{{ 24k = 72 }}}
{{{ k = 3 }}}
The roots are:
{{{ -1/2 + (1/2)*i }}}
{{{ -1/2 - (1/2)*i }}}
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