Brian's average mark on three mathematics tests was 78. The mark on the first test was 86. His average for the first two tests was three more than the mark on the third test. What marks did Brian get on the second and third tests?

The marks that Brian get on the second test is 72

The marks that Brian get on the third test is 76

Step-by-step explanation:

We have a question with a word problem

The parameters given are;

Average mark on three mathematics test = 78

The mark received on the first test, M₁ = 86

The average for the first two test = 3 + Mark received on the third test M₃

Let M₂ = The mark received on the second test

Therefore, we have three equations with three equations as follows;

(M₁ + M₂)/2 = 3 + M₃   ...(1)

(M₁ + M₂ + M₃)/3 = 78(2)

M₁ = 86(3)

Which gives;

(86 + M₂)/2 = 3 + M₃   ...(4)

(86 + M₂ + M₃)/3 = 78(5)

Making M₂ the subject of equation (4) gives;

M₂ = 2 × (3 + M₃) - 86

Replacing the value of M₂ in equation (5) gives;

(86 + (2 × (3 + M₃) - 86) + M₃)/3 = 78

(86 + (6 + 2·M₃) - 86) + M₃)/3 = 78

(6 + 2·M₃ + M₃)/3 = 78

6 + 3·M₃ = 78×3 = 234

3·M₃ = 234 - 6 =   228

M₃ = 228/3 = 76

From  M₂ = 2 × (3 + M₃) - 86 = 6 + 2·M₃ - 86 = 2·M₃ - 80

We have;

M₂ =  2×76 - 80 = 72

Therefore, Brian got 72 on the second test and 76 on the third test.

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

answer: -1

step-by-step explanation: