Alan

A cubic polynomial has three roots.  A quartic polynomial has four.

This should help (it uses the chain rule):

All the sides of a triangle have integer length. The perimeter of the triangle is 50 and the triangle is isosceles. How many such non-congruent triangles are there?

Let s be the length of two of the sides and b be the length of the third side.

We must have \(50=2s+b\) and \(b<2s\)

Try values of s from 1 upwards until the second expression above is violated, remembering that b must be an integer.

This should help (Note: distances scaled by a factor of 10):

A sphere passes through all four vertices of one face of a unit cube, and is tangent to the opposite face of the cube. Find the radius of the sphere.

Let b = initial number of blue pens and r = initial number of red pens.

Form the following two equations and solve them for r and b:

\(\frac{b}{r}=\frac{4}{5}\\ \frac{b-280}{r-280}=\frac{9}{20}\)

Look at https://web2.0calc.com/questions/probability-problem_24#r2.  Use the same method, suitably modifying the numbers.

Have a look at this video:  https://www.youtube.com/watch?v=l69YjkuUym0

Here's a starter:

As follows:

The question specifies that \(k \ne 1\) 

x = 1 is certainly a solution, but there are two other real number solutions to the original cubic equation!

(1/2) * 11 * h = (1/2) * (h - 5) * 16  ?????!!!!!