Let b1 = and b = E: {61, 62 is a basis for R?. Let T : R? → R² be a linear transformation such that The set B = T(51) = 851 + 2b, and T(52) = 2b1 + 862. (a) The matrix of T relative to the basis B is (T]B =

the center and foci of the followin 4x^(2)+8y^(2)-16x+32y+32=0

Find the value of c such that the exp k^(2)+13k+c

show work

goes through the points (-3,2) and (8,2)

I have to find a and b when the gcd of these two natural numbers is 7, and a^2 + b^2=490

express (7,-3,1)  in linear combination of (5,3,9), (-3,2,0) and (6,0,-8)

A tunnel is being constructed through a mountain range where the tunnel is being constructed from both ends at the same time. One end of tunnel starts at the point A (80, -287, 106) and is constructed in a straight line in the direction 2î - 8j+ 3 k whilst the other end of tunnel starts at the point B (-395,263, -89) and is constructed in a straight line in the direction 9i - 6j +2k. The starting points and the direction vectors of both ends of the tunnel are measured in metres. Verify that the tunnels being constructed from both ends meets by first finding the equations of both tunnels and then finding their intersection point P.

Consider the polynomial:$$f=2 X^{4}+a X^{3}+3 X^{2}+b X+c \in \mathbf{R}[X]$$with roots $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{X}_{3}, \mathrm{X}_{4} \in \mathbb{C}$.a) Calculate:$$\sum_{1 \leq i<j \leq 4}\left(x_{i}-x_{j}\right)^{2}$$and then prove that for | a | <4  the polynomial has at most two real roots. Please check the attached picture for details. I need complete solution with explanation please.

Consider the polynomial:$$f=2 X^{4}+a X^{3}+3 X^{2}+b X+c \in \mathbf{R}[X]$$with roots $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{X}_{3}, \mathrm{X}_{4} \in \mathbb{C}$.a) Calculate:$$\sum_{1 \leq i<j \leq 4}\left(x_{i}-x_{j}\right)^{2}$$and then prove that for | a | <4  the polynomial has at most two real roots. b) For  | a | = 4  determine b and c so that f has all roots real. c) For  | a |> 4  prove that there is b and c in $\mathbb{R}$  such that f has all roots real. Please check the attached picture for details. I need complete solution with explanation please. All three subpoints are subpoints for the same problem and they are conected between them, but if it is not possible to solve them all please solve what you can, thank you in advance.