Multiplying numbers with the same base (for this problem, the base is 2) can be simplified by adding the powers.
1/4 + 1/8 + 1/16 + 1/32 ... = 1/2
The fractions form an infinite geometric series whose first term is 1/4 and whose common ratio is 1/2.
Sum = (1/4) / [ 1 - 1/2 ] = (1/4) / (1/2) = 1/2
So, the product can be written as 21/2 = sqrt( 2 )
This means that a = 2 and b = 2.
a + b = 2 + 2 = 4
y = -2x2 + 8x - 15
Move the '-15' term to the other side by adding 15 to both sides:
y + 15 = -2x2 + 8x
Factor out the '-2' term:
y + 15 = -2(x2 - 4x)
Complete the square of x2 - 4x ---> x2 - 4x + 4
Notice that the '4' will be on the inside of the factor, so really, it is a '-8':
y = 15 - 8 = -2(x2 - 4x + 4)
y + 7 = -2(x - 2)2
The vertex occurs at (2, -7)
If these five letters are each intended for the five persons (one per person), then, with random distribution, it is impossible for exactly four of the persons to get the correct letter. If each of the four persons get the letters intended for him/her, then the fifth person will get the letter intended for him/her.
The probability is zero.
Multiplying this out the long way is tedious, put will give you the correct answer.
You can use the shortcut: (x + y)6 = x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6
where x = 3/5 and y = 4/5·i. (You can get the coefficients by using either combinations, or Pascal's triangle.)
Finally, find the absolute value of the answer.
Since one root is 2 - i another root will be 2 + i.
Since one root is 2 + sqrt(3) another root will be 2 - sqrt(3).
Written as factors: [ x - (2 - i) ] · [ x - (2 + i) ] · [ x - (2 + sqrt(3)) ] · [ x - (2 - sqrt(3)) ] = 0
Multiply this out and select the constant term.
However, since this answer can be multiplied by any non-zero constant and still have the same roots, any non-zero constant will work.