Question

What two numbers has sum equal to 9/4 and the product equal to 3/4?

Answer 1

`9/8+sqrt(33)/8" "` and `" "9/8-sqrt(33)/8`

Explanation:

Notice that given two numbers `a` and `b`:

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

So the two numbers we want will be the zeros of the quadratic polynomial:

`f(x) = x^2-9/4x+3/4`

So we find:

`0 = 64f(x)`

`color(white)(0) = 64(x^2-9/4x+3/4)`

`color(white)(0) = 64x^2-144x+48`

`color(white)(0) = (8x)^2-2(8x)(9)+81-33`

`color(white)(0) = (8x-9)^2-(sqrt(33))^2`

`color(white)(0) = ((8x-9)-sqrt(33))((8x-9)+sqrt(33))`

`color(white)(0) = (8x-9-sqrt(33))(8x-9+sqrt(33))`

Hence:

`x = 1/8(9+-sqrt(33)) = 9/8+-sqrt(33)/8`

That is, the two numbers are:

`9/8+sqrt(33)/8" "` and `" "9/8-sqrt(33)/8`