Respuesta :
Answer: a) k >4.08
b) k = 4.08
c) k<4.08
Step-by-step explanation:
Since we have given that
[tex]f(x)=-3x^2+7x-k[/tex]
a) For what values of k will the function have no zeros?
It mean it has no real zeroes i.e. Discriminant < 0
As we know that
[tex]D=b^2-4ac[/tex]
Here, a =-3
b = 7
c = -k
So, it becomes,
[tex]D<0\\\\b^2-4ac<0\\\\7^2-4\times -3\times -k<0\\\\49-12k<0\\\\-12k<-49\\\\k>\dfrac{49}{12}\\\\k>4.08[/tex]
b) For what values of k will the function have one zero?
It means it has one real root i.e equal roots.
So, in this case, D = 0
So, it becomes,
[tex]D=b^2-4ac=0\\\\D=7^2-4\times -3\times -k=0\\\\49-12k=0\\\\49=12k\\\\k=\dfrac{49}{12}\\\\k=4.08[/tex]
c) For what values of k will the function have two zeros?
It means it has two real roots.
In this case, D>0
So, it becomes,
[tex]D=7^2-4\times -3\times -k>0\\\\49-12k>0\\\\-12k>-49\\\\12k<49\\\\k<4.08[/tex]
Hence, a) k >4.08
b) k = 4.08
c) k<4.08
