Question

How do you solve `x^2-4x-2=0` by using the Quadratic Formula?

Answer 1

The solutions are:
`color(blue)(x=2+sqrt6`
`color(blue)(x=2-sqrt6`

Explanation:

The equation: `x^2-4x-2` is of the form `color(blue)(ax^2+bx+c=0` where:
`a=1, b=-4, c=-2`

The Discriminant is given by:

`Delta=b^2-4*a*c`

`= (-4)^2 - (4)* (1)*(-2)`
`= 16+8=24`

As `Delta>0` there are two solutions,

The solutions are found using the formula
`x=(-b+-sqrtDelta)/(2*a)`

`x = (-(-4)+-sqrt(24))/(2*1) = (4+-sqrt(24))/2`

`(color(blue)(sqrt24 = sqrt(2*2*2*3) = 2sqrt6)`

So,
`x= (4+-2sqrt(6))/2`

Taking `2` outside the bracket as it is common to both terms of the numerator

`x= (cancel2(2+-sqrt(6)))/cancel2`
the solutions are:
`color(blue)(x=2+sqrt6`
`color(blue)(x=2-sqrt6`