1. Number of roots

In a question, I was given a graph which has a cubic curve, passing through x=-2 and touches 0.

The question asks me to explain how the number of roots of the equation f(x)=k(x+2) depends on k.

Sorry,I haven't had any maths program so I'm unable to draw the graph here. So I'm not expecting any full solutions but some advice on solving such problem.

My attempt: Since the curve touches zero, obviously it has two real and equal roots which are zero.

Hence, I assume that f(x)=k=x^2

if K>0, the cup-shaped curve would be heading upwards. (If the cup-shaped curve has centre of (0,0), it intersects with the curve f(x)=k(x+2), having two equal roots... )

if K=0, the means f(x)=0, which means the number of roots are two equal and real roots.

if K<0, the cup-shaped curve would be heading downwards, so to calculate the number of roots, I should take into account all possible intersections between this cup-shaped curve with the curve f(x)=k(x+2) right?

Thanks for your advice!! Sorry if my writing is confused for you
Looking forward to hearing from you soon.

2. Originally Posted by ose90
In a question, I was given a graph which has a cubic curve, passing through x=-2 and touches 0.

The question asks me to explain how the number of roots of the equation f(x)=k(x+2) depends on k.

Sorry,I haven't had any maths program so I'm unable to draw the graph here. So I'm not expecting any full solutions but some advice on solving such problem.

My attempt: Since the curve touches zero, obviously it has two real and equal roots which are zero.

Hence, I assume that f(x)=k=x^2

if K>0, the cup-shaped curve would be heading upwards. (If the cup-shaped curve has centre of (0,0), it intersects with the curve f(x)=k(x+2), having two equal roots... )

if K=0, the means f(x)=0, which means the number of roots are two equal and real roots.

if K<0, the cup-shaped curve would be heading downwards, so to calculate the number of roots, I should take into account all possible intersections between this cup-shaped curve with the curve f(x)=k(x+2) right?

Thanks for your advice!! Sorry if my writing is confused for you
Looking forward to hearing from you soon.
In my opinion this question asks something different:

According to the wording k must be a function of x. (If k is a constant then f is a linear function and not a cubic one).

If D denotes the domain of f then

- you only get one real root if $k(x) \neq 0$ for all $x \in D$

- you get two real roots of f if k(x) is a perfect square and $x \neq -2$.

- you get three real roots if k(x) has two real roots which are different from x = -2.

3. Originally Posted by earboth
In my opinion this question asks something different:

According to the wording k must be a function of x. (If k is a constant then f is a linear function and not a cubic one).

If D denotes the domain of f then

- you only get one real root if $k(x) \neq 0$ for all $x \in D$

- you get two real roots of f if k(x) is a perfect square and $x \neq -2$.

- you get three real roots if k(x) has two real roots which are different from x = -2.
Thanks alot for your help! I've misunderstood the question