1. 1/x^2+6/x+9 =0

multiply all terms by x^2
1+6x+9x^2 = 0

factorizing, 9x^2 = 3x*3x
9x^2 + 3x + 3x + 1 = 0

(3x+1)^2 = 0

x = -1/3


2. x^2(x^4+2*x^2-35) = 0

x^2 = 0 gives x =0 as one root.

solving for the other polynomial, you'll see
-35*x^4 = 7*x^2 * -5*x^2

so, (x^2-5)(x^2+7) =0

x^2 =5 gives x = +/- rt. 5

so , 3 real factors.

I wish I could post it better but typing out exponents & mathematical expressions in text are just painful, so I am trying to shorten it here.

3. A sphere with diameter 6 is inscribed into a squared box. What is the greatest stick that we can put between the sphere and the box?


I have attached a 2D view of a sphere in a square box. We can assume that the edges of the square act as tangents to the circle. Then maximum length of a stick that can be inserted between the sphere and the box will be equal to the radius of the circle = 3.

Corner right triangle that will be formed will have sides of length 3 each. But we cannot insert a stick of length equal to that of a hypotenuse because it will be blocked by the sphere. Hence maximum length possible is that of the edge of the triangle = radius of the circle = 3.

Caps, please post one question per post. This helps to keep everything organized and focused on one topic.

Also, make sure you post each question in the appropriate section of the forum.