Final answer:
Liam's age can be determined by setting up and solving a quadratic equation based on the given conditions. Chloe's age is represented as x, and Liam's age is x + 1. After solving, we find that Liam is 9 years old.
Step-by-step explanation:
We are given that Liam's age and Chloe's age are consecutive integers and that Liam is older than Chloe. Let's denote Chloe's age as x and thus Liam's age is x + 1. The problem tells us that the sum of the square of Liam's age and 5 times Chloe's age equals 121. This leads to the following equation:
(x + 1)^2 + 5x = 121
Now we can solve this quadratic equation:
x^2 + 2x + 1 + 5x = 121
x^2 + 7x + 1 = 121
x^2 + 7x + 1 - 121 = 0
x^2 + 7x - 120 = 0
To solve this, we look for factors of -120 that add up to 7. The factors -8 and 15 come to mind.
x^2 - 8x + 15x - 120 = 0
(x - 8)(x + 15) = 0
This gives us two possible solutions for x:
x = 8 or x = -15
We can disregard -15, as ages cannot be negative. Therefore, Chloe is 8 years old and Liam, being one year older, is 9 years old. So, the correct answer is D. 9.