maths

1)

Compute the value of x such that $\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots.$

2)

The graph of the parabola x = 2y^2 - 6y + 3 has an x-intercept (a,0) and two y-intercepts (0,b) and (0,c). Find a + b + c.

3)

Circles with centers of (2,2) and (17,10) are both tangent to the x-axis. What is the distance between the closest points of the two circles?

4)

Given the function y=x^2+10x+21, what is the least possible value of y?

5)

What real values of x are not in the domain of $f(x)=\frac{1}{|x^2+3x-4|+|x^2+9x+20|}?$

6)

Define the operation $a\nabla b = b^a - 2$. What is the value of $(2\nabla 3) \nabla 2$?