1)let k and w be two consecutive integers such that k<x<w. If log base 7 of 143 = x, find the value of k+w

Question

2) if 7 and -1 are two of the solutiosn for x in the equation 2x^3 +kx^2 -44x+w=0, find the value of k+w

3) from an ordinary deck of 52 cards, two cards are selected at random (without replacement). find the probability that both cards were hearts. express your answer as a common fraction reduced to lowest terms.

Answer 1

1. if log₇ 143 = x
then 7^x = 143
but 7^2 = 49 and 7^3 = 343
so k=2 and w = 3
then k+w = 5

2. let f(x) = 2x^3 + kx^2 - 44x + w
if 7 is a solution then f(7) = 0
2(343) + 49k - 308 + w = 0
49k + w = -378

if -1 is a solution, then f(-1) = 0
2(-1) + k + 44 + w = 0
k + w = -42
subtract them:
48k = -336
k = -7
in k+w=-42, -7+w = -42 ---> w = -35

and k+w = -42

3) prob(2 hearts) = (13/52)(12/51) = 1/17

Answer 2

thank you!