Under what circumstances would a quadratic inequality have a solution set that is a closed interval?

Question

Can someone help me with this? The teacher wasnt in class and wants us to do this for homework and I am lost. Anything would be great...Thank you very much

under what circumstances would a quadratic inequality have a solution set that is a closed interval? Under what circumstances would a quadratic inequality have an empty solution set? Make an example for each situation.

Answer 1

a quadratic inequality of the form

-ax^2 + bx + c => 0,

where b^2 - 4ac => 0 satisfies your conditions.

I put b^2 - 4ac => 0 because that ensures that the function

f(x) = -ax^2 + bx + c

has real roots.

the graph of f(x) = -ax^2 + bx + c opens downward. so, f(x) => 0 only for values of x between its roots. so, the solutions of

-ax^2 + bx + c => 0 lie between its roots, making them a closed interval.

another form of quadratic inequality

ax^2 + bx + c <= 0 , where b^2 - 4ac => 0 also has a closed interval solution set.

2) the quadratic inequality,

ax^2 + bx + c <= 0 where b^2 - 4ac < 0 has an empty solution set.

under the same conditions, -ax^2 + bx + c => 0 also has an empty solution set.