For this case we have the following quadratic equation:
![3x ^ 2 + 7x + 4 = 0](https://tex.z-dn.net/?f=3x%20%5E%202%20%2B%207x%20%2B%204%20%3D%200)
Where:
![a = 3\\b = 7\\c = 4](https://tex.z-dn.net/?f=a%20%3D%203%5C%5Cb%20%3D%207%5C%5Cc%20%3D%204)
According to the quadratic formula we have:
![x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2 (a)}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%20%7B-b%20%5Cpm%20%5Csqrt%20%7Bb%20%5E%202-4%20%28a%29%20%28c%29%7D%7D%20%7B2%20%28a%29%7D)
Substituting:
![x = \frac {-7 \pm \sqrt {7 ^ 2-4 (3) (4)}} {2 (3)}\\x = \frac {-7 \pm \sqrt {49-48}} {6}\\x = \frac {-7 \pm \sqrt {1}} {6}\\x = \frac {-7 \pm1} {6}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%20%7B-7%20%5Cpm%20%5Csqrt%20%7B7%20%5E%202-4%20%283%29%20%284%29%7D%7D%20%7B2%20%283%29%7D%5C%5Cx%20%3D%20%5Cfrac%20%7B-7%20%5Cpm%20%5Csqrt%20%7B49-48%7D%7D%20%7B6%7D%5C%5Cx%20%3D%20%5Cfrac%20%7B-7%20%5Cpm%20%5Csqrt%20%7B1%7D%7D%20%7B6%7D%5C%5Cx%20%3D%20%5Cfrac%20%7B-7%20%5Cpm1%7D%20%7B6%7D)
We have two roots:
![x_ {1} = \frac {-7 + 1} {6} = \frac {-6} {6} = - 1\\x_ {2} = \frac {-7-1} {6} = \frac {-8} {6} = - \frac {4} {3}](https://tex.z-dn.net/?f=x_%20%7B1%7D%20%3D%20%5Cfrac%20%7B-7%20%2B%201%7D%20%7B6%7D%20%3D%20%5Cfrac%20%7B-6%7D%20%7B6%7D%20%3D%20-%201%5C%5Cx_%20%7B2%7D%20%3D%20%5Cfrac%20%7B-7-1%7D%20%7B6%7D%20%3D%20%5Cfrac%20%7B-8%7D%20%7B6%7D%20%3D%20-%20%5Cfrac%20%7B4%7D%20%7B3%7D)
Answer:
![x_ {1} = - 1\\x_ {2} = - \frac {4} {3}](https://tex.z-dn.net/?f=x_%20%7B1%7D%20%3D%20-%201%5C%5Cx_%20%7B2%7D%20%3D%20-%20%5Cfrac%20%7B4%7D%20%7B3%7D)
The probability you're asked to find is
![\mathbb P(A\cup B)-\mathbb P(A\cap B)](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Ccup%20B%29-%5Cmathbb%20P%28A%5Ccap%20B%29)
where
![A\cup B](https://tex.z-dn.net/?f=A%5Ccup%20B)
is the event that either event occurs (A, B, or both), and
![A\cap B](https://tex.z-dn.net/?f=A%5Ccap%20B)
is the event that both events occur.
Recall that
![\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Ccup%20B%29%3D%5Cmathbb%20P%28A%29%2B%5Cmathbb%20P%28B%29-%5Cmathbb%20P%28A%5Ccap%20B%29)
which means
![\mathbb P(A\cup B)-\mathbb P(A\cap B)=\mathbb P(A)+\mathbb P(B)-2\mathbb P(A\cap B)](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Ccup%20B%29-%5Cmathbb%20P%28A%5Ccap%20B%29%3D%5Cmathbb%20P%28A%29%2B%5Cmathbb%20P%28B%29-2%5Cmathbb%20P%28A%5Ccap%20B%29)
You're told that
![\mathbb P(A)=0.5](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%29%3D0.5)
,
![\mathbb P(B)=0.7](https://tex.z-dn.net/?f=%5Cmathbb%20P%28B%29%3D0.7)
, and (if I'm reading the diagram correctly)
![\mathbb P(A\cap B)=0.3](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Ccap%20B%29%3D0.3)
.
So,
![\mathbb P(A\cup B)-\mathbb P(A\cap B)=0.5+0.7-2\times0.3=0.6](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Ccup%20B%29-%5Cmathbb%20P%28A%5Ccap%20B%29%3D0.5%2B0.7-2%5Ctimes0.3%3D0.6)
Another way of seeing this is that the event A consists of the regions "A not B" and "A and B". So the probability that "A not B" occurs is
![\mathbb P(A\setminus B)=\mathbb P(A)-\mathbb P(A\cap B)=0.5-0.3=0.2](https://tex.z-dn.net/?f=%5Cmathbb%20P%28A%5Csetminus%20B%29%3D%5Cmathbb%20P%28A%29-%5Cmathbb%20P%28A%5Ccap%20B%29%3D0.5-0.3%3D0.2)
Similarly, B consists of "B not A" and "A and B", so you have
![\mathbb P(B\setminus A)=\mathbb P(B)-\mathbb P(A\cap B)=0.7-0.3=0.4](https://tex.z-dn.net/?f=%5Cmathbb%20P%28B%5Csetminus%20A%29%3D%5Cmathbb%20P%28B%29-%5Cmathbb%20P%28A%5Ccap%20B%29%3D0.7-0.3%3D0.4)
So the probability that A or B, but not both, occur is
Well, the ratio is 24:6 they want you to change 6 to 1 so that means we need to divide 24 by 6 to get 4 so, 4 is your answer.
X - green marbles
X + 3 - blue marbles
Total = 23
X + X + 3 = 23
2X + 3 = 23
2X = 23 - 3
2X = 20
X = 10
Jerry has 10 green marbles. He also has 13 blue Marbles. To give a total of 23.
The equation would be y=1/3x. The slope rises 1 unit for every 3 units it moves right.