Question

The sum of two numbers is 10 and the sum of their square is 68. Find the numbers

Answer 1

2 and 8

Explanation:

Using the given information, we can represent the numbers (a and b) in a system of equations:

eq 1. "the sum of two numbers is 10"

`a + b = 10`

eq 2. "the sum of their square is 68"

`a^2 + b^2 = 68`

Then, we can solve for each number using substitution.

`\begin{cases} a+ b = 10 \\ a^2 + b^2 = 68\end{cases}`

We can first solve for a in terms of b in the first equation.

`a+b=10`

`a=10-b`

Then, we can substitute this a-value into the second equation and solve for b.

`a^2 + b^2 = 68`

↓ substituting the solution for a in terms of b

`(10-b)^2 + b^2 = 68`

↓ expanding the binomial ...
`(c - d)^2 = c^2 -2cd + d^2`

`(10^2-2(10b)+b^2) + b^2 = 68`

↓ simplifying

`2b^2-20b+100 = 68`

↓ subtracting 68 from both sides

`2b^2-20b+32 = 0`

↓ dividing both sides by 2

`b^2 - 10b + 16 = 0`

↓ factoring ...
`(-2) + (-8) = -10` and
`(-2)(-8) = 16`

`(b-2)(b-8) = 0`

↓ solving for b ... if
`AB=0`, then
`A=0` or
`B=0`

b = 2 or b = 8

Now, we can try plugging each b-value into the solution for a in terms of b, then solve for a.

`a=10-b`

`a=10-2`

`\boxed{a=8}`

OR

`a=10-8`

`\boxed{a=2}`

So, the two numbers are 2 and 8.