20 !

does the equation x2 - 4x + y2 = -3 intersect the x-axis?

Yes, twice

Step-by-step explanation:

The equation will intersect the x-axis when y = 0, so we have

x² - 4x = -3    now solve this quadratic for x...

  x² - 4x + 3 = 0    

factor...

(x - 3)(x - 1) = 0,

so at x = 1 and x = 3, the function crosses the x-axis

See the graph below

Yes, because the center is on the x-axis, the circle will intersect 2 times, at x=1 and at x=3

answer: slight chance that it will happen

step-by-step explanation:

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}