Darrel is 8 years older than bryce. if the product of their ages is 153, how old is bryce?

Let D = Darrel's age and let B = Bryce's age.

The first statement is that Darrel is 9 years older. So, Bryce's age and 9 makes Darrel's age. This means B + 9 = D.

The second statement is that their product is 153. Thus, B * D = 153

We take these two equations made and put them together. Call them 1 and 2 to keep them straight.

1: B + 8 = D

2: BD = 153

Equation 1 has D on one side of the equals, and not D on the other. We can substitute it into equation 2.

B (B + 8) = 153 taking equation 1 and putting it into equation 2

B² + 8B = 153 distributing the b on the left side

B² + 8B - 153 = 0 set the equation equal to zero

We set the equation equal to zero so we can use the Zero Product Property and factoring. We seek a pair of numbers whose product is 153 and whose sum is 8. Because 153 ÷ 3 = 51 and 51 ÷ 3 = 17, there are only three pairs of factors: 1 and 153, 3 and 51, 9 and 17. Of them, 9 and 17 would add to 8 if one was negative. We have a positive 8 above, so choose the smaller one to be negative.

(B + 17) (B - 9) = 0 by factoring

B + 17 = 0 or B - 9 = 0 by the zero product property

B = -17 or B = 9 Get B alone in each equation

So either Bryce is -17 years old or 9 years old. Ages cannot be negative, so throw out B = -17 and we use B = 9.

Now we go back to the first statement that Darrel is eight years older than Bryce. We found that Bryce is 9, so we add 8 to get Darrel's age, which is 17.

Thus, Bryce is 9 years old and Darrel is 17 years old.

(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))