Question 993457
.
For what value of B does the function F(x) = {{{-3x^2+Bx-4}}}
have only 1 x-intercept. Find all possible values of B and leave your answer in exact form. 
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It may happen if and only if two real roots of the quadratic polynomial  F(x) = {{{-3x^2+Bx-4}}}  merge into one 

real root of multiplicity  2.  It is the case when the plot of the quadratic polynomial touches  x-axis.


In turn,  a necessary and a sufficient condition for it is vanishing the discriminant of the polynomial,  which is 


{{{B^2}}} - 4*(-3)*(-4) = 0,     or


{{{B^2}}} = 16*3.


It gives   B = {{{4*sqrt(3)}}}   and/or   B = - {{{4*sqrt(3)}}}. 


At these values of B   F(x) = {{{-(sqrt(3)*x - 2)^2}}}   or   F(x) = {{{-(sqrt(3)*x + 2)^2}}}   correspondingly.


<U>Answer</U>. &nbsp;&nbsp;B = {{{4*sqrt(3)}}} &nbsp;&nbsp;and/or &nbsp;&nbsp;B = - {{{4*sqrt(3)}}}.

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -5.5, 5.5, -10.5, 5.5,
          -(sqrt(3)*x - 2)^2,
          -(sqrt(3)*x + 2)^2
)}}}


Plots {{{-(sqrt(3)*x - 2)^2}}} (in red) and {{{-(sqrt(3)*x + 2)^2}}} (in green)

  </TD>
  </TR>
</TABLE>