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.
