Question

Find the exact solutions and simplify as much as possible: 3x^2+1=6x

Answer 1

`(3+ √(6) )/(3)` and
`(3- √(6) )/(3)`

Explanation:

This is a quadratic equation, so we start by rearranging its terms so we can apply the quadratic formula:

`ax^2+bx+c=0` has two possible solutions given by:

`x=(-b+/- √(b^2-4ac) )/(2a)`

So we rearrange the terms in the given quadratic equation bringing all terms to the left hhand side of the equal sign:

`3x^2+1=6x\\3x^2-6x+1=0`

The solutions will then be:

`x=(6+/- √(36-4(3)(1)) )/(2(3))=(6+/- √(36-12) )/(6)=(6+/- √(24) )/(6)=(6+/- √(4*6) )/(6)=(6+/- 2√(6) )/(6)=(3+/- √(6) )/(3)`

Therefore the two possible answer are:
`(3+ √(6) )/(3)` and
`(3- √(6) )/(3)`