Question 1029915
For there to be just 1 solution, the part
of the quadratic formula called the 
discriminant must equal zero.
The discriminant is:
{{{ b^2 - 4*a*c }}}
The quadratic formula is:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
when you are given the general form:
{{{ f(x) = a*x^2 + b*x + c }}}
this is how you find a, b, and c
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So, for 1 solution:
{{{ b^2 - 4*a*c = 0 }}}
{{{ b^2 = 4*a*c }}}
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You are given:
{{{ p*x^2 - 12x + 4 = 0 }}}
{{{ b^2 = (-12)^2
{{{ b^2 = 144 }}}
and
{{{ 4*a*c = 4*p*4 }}}
and
{{{ 144 = 4*p*4 }}}
{{{ p = 9 }}}
If {{{ p = 9 }}}, there is only 1 solution
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check the answer
{{{ 9x^2  - 12x + 4 = 0 }}}
{{{ x = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ x = ( -(-12) +- sqrt( (-12)^2 - 4*9*4 )) / (2*9) }}}
{{{ x = ( 12 +- sqrt( 144 - 144 )) / 18 }}}
{{{ x = ( 12 +- 0 ) / 18 }}}
{{{ x = 2/3 }}}
So, there is only 1 solution