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Probability is the likelihood that an event will occur...like 1/2 for a coin, or 1/6 for a die. ☺️

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-15 ≤ 5x < 20

-15/5 ≤ 5x/5 < 20/5

-3 ≤ x < 4

[-3, 4)

Square roots of negative numbers are called imaginary numbers.

Therefore, √-18, √-3, -√-7 are all imaginary numbers.

The letter "i" is used to represent √-1   ---

so  √-7 =  √-1√7  =  i√7.

Law of Cosines:  c²  =  a² + b² -2abcos(C)

For this problem:  c = AB     a  =  BC = 700     b = AC = 650

--->                  c²  =  700² + 650² - 2(700)(650)cos(121.1°)

--->                  c²  =  1382545

--->                    c  =  1176.  

x² + 8x + 15  =  -1

Get all the terms to one side (add 1 to both sides):

x² + 8x + 16  =  0

To factor:  find two numbers that:  when added together give you 8 and when multiplied give you 16.

(x + 4)(x + 4)  =  0

When does  x + 4  =  0?  

When x  =  -4!

(1/2) the time he rolls a 6:  (1/2) x 6  =  3.

(1/10) the time he rolls a 1:  (1/10) x 1  =  1/10

(1/10) the time he rolls a 2:  (1/10) x 2  =  2/10

(1/10) the time he rolls a 3:  (1/10) x 3  =  3/10

(1/10) the time he rolls a 4:  (1/10) x 4  =  4/10

(1/10) the time he rolls a 5:  (1/10) x 5  =  5/10

Sum those values:  3 + 1/10 + 2/10 + 3/10 + 4/10 + 5/10  =  4.5

Expected value is 4.5

The size of a television set is measured along its diagonal.

The length, width, and diagonal form a right triangle, so we can use the Pythagorean Theorem:

(length)²  +  (width)²  =  (diagonal)²

(59)²  +  (width)²  =  (120)²

3481  +  (width)²  =  14400

(width)²  = 10919

width  =  104.5 in. (approximately)

The area of a square with side length x is x².

The area of a circle with radius (x - 1)  is Π(x - 1)²  = Π(x - 1)(x - 1)  =  Π(x² - 2x + 1)  =  Πx² - 2Πx + Π

Since they are equal:  x²  =  Πx² - 2Πx + Π

Getting all terms to one side:  0  =  Πx² - x² - 2Πx + Π

Factoring:                             0  =  (Π - 1)x² - 2Πx + Π

Using the quadratic formula with a = (Π - 1), b = -2Π, and c = Π

x  =  [2Π ± √( (-2Π)² - 4(Π - 1)(Π) ) ] / [ 2(Π - 1) ]

x  =  [2Π ± √( 4Π² - 4Π² + 4Π ) ] / [ 2(Π - 1) ]

x  =  [2Π ± √(4Π) ] / [ 2(Π - 1) ]

x  =  [2Π ± 2√(Π) ] / [ 2(Π - 1) ]

x  =  [Π ± √(Π) ] / [ Π - 1 ]

So, either     x  =  0     or     x  =  [Π + √(Π) ] / [ Π - 1 ]

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