Question

Consider the function f(x)=-3x^2 +7x -k. [3 Marks] a) For what values of k will the function have no zeros? b) For what values of k will the function have one zero? c) For what values of k will the function have two zeros?

Answer 1

Answer: a) k >4.08

b) k = 4.08

c) k<4.08

Explanation:

Since we have given that

`f(x)=-3x^2+7x-k`

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

`D=b^2-4ac`

Here, a =-3

b = 7

c = -k

So, it becomes,

`D<0\\\\b^2-4ac<0\\\\7^2-4* -3* -k<0\\\\49-12k<0\\\\-12k<-49\\\\k>(49)/(12)\\\\k>4.08`

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,

`D=b^2-4ac=0\\\\D=7^2-4* -3* -k=0\\\\49-12k=0\\\\49=12k\\\\k=(49)/(12)\\\\k=4.08`

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,

`D=7^2-4* -3* -k>0\\\\49-12k>0\\\\-12k>-49\\\\12k<49\\\\k<4.08`

Hence, a) k >4.08

b) k = 4.08

c) k<4.08