Question

If ab = 9 and a² + b² = 81, then (a+b)² = ?​

Answer 1

99

Explanation:

Through some clever algebra, we can combine our first two equations using the fact that

`(a+b)^2=a^2+2ab+b^2`

We have the a² and the b² in the equation
`a^2+b^2=81`, and we almost have a 2ab in the equation
`ab=9`. To get that 2ab, we can simply double both sides of the first equation to get
`2ab=18`. From there, we'll add the first two equations together, giving us the summed equation

`a^2+b^2+2ab=81+18\\a^2+2ab+b^2=99`

And since
`(a+b)^2=a^2+2ab+b^2`, we can equivalently say that

`(a+b)^2=99`