Given composition xoy=2(x-3)(y-3)+3, solve 5^xo5^x=11?

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to replace `5^x` for `x` and `y` in the given law of composition, such that:

`5^xo5^x = 2(5^x - 3)(5^x - 3) + 3`

Replacing `2(5^x - 3)(5^x - 3) + 3` for `5^xo5^x` in equation, yields:

`2(5^x - 3)(5^x - 3) + 3 = 11 => 2(5^x - 3)(5^x - 3) = 11 - 3`

`2(5^x - 3)(5^x - 3) = 8 => (5^x - 3)(5^x - 3) = 4 => (5^x - 3)^2 = 4 => (5^x - 3)^2 - 4 = 0`

You need to convert the difference of squares into a product, such that:

`(5^x - 3 - 2)(5^x - 3 + 2) = 0 => (5^x - 5)(5^x - 1) = 0`

Using the zero product rule yields:

`{(5^x - 5 = 0),(5^x - 1 = 0):} => {(5^x = 5),(5^x = 1):}`

`{(5^x = 5^1),(5^x = 5^0):} => {(x = 1),(x = 0):}`

Hence, evaluating the solutions to the given equation, using the given law of composition, `5^xo5^x = 11` , yields `x = 0, x = 1.`

We’ve answered 315,731 questions. We can answer yours, too.

Ask a question