When you add the square of Thomas's age to Lauren's age the total is 62. When you add the square of Lauren's age to Thomas's age the total is 176. How old are Thomas and Lauren?

Question

When you add the square of Thomas's age to Lauren's age the total is 62. When you add the square of Lauren's age to Thomas's age the total is 176. How old are Thomas and Lauren?

anonymous · Answer

Let T=Thomas
L=Lauren
$$T^2+L = 62$$
$$T+L^2=176$$

Now that you have a system of equations, you can solve for both variables. I suggest doing substitution, followed by factoring to find the roots.

anonymous · Answer

T=7 and L=13 if you want to check your answer. I have to leave so i can't guide you to reach it.

anonymous · Answer

Could you show your work please.

anonymous · Answer

Hey @2le

anonymous · Answer

@2le please.

anonymous · Answer

$$L=62-T^2$$
Plug that into the next equation
$$T+L^2=T+(62-T^2)^2=176$$
$$T+62^2+T^4-2*62*T^2=176$$
$$T^4-2*62*T^2+T+62^2-176=T^4-124*T^2+T+3668=0$$
$$(T-7)(T^3+7T^2-75T+524)=0$$
That's a very difficult factor to guess. You can list the prime factors of 3668 to find the potential roots (2*2*7*131=3668) Of these, you have several sets of realistic ages. You can have 2, 4, 7, 13, 14, 26, 28, ...once you get to 131 though, you can start ruling it out. 
Going to back to your original equation, L=62-T^2
Let's look at a few of these possible answers. If T=13, L=62-(13)^2=62-169=-107. How unrealistic, he's got a negative age! That's 107 years before he was born. That tells you that that age is too high.
Let's try T=7. L=62-(7)^2=62-49=13 Now that, I can believe.
Let's try plugging that 7 into the other equation. 7 + L^2 = 176. L^2=176-7=169 => L = sqrt(169)=13
If you try any other possible roots, it'll show negative or ridiculous ages that doesn't work.