Question

How do you solve `2x^2-6x+3>=0` using a sign chart?

Answer 1

The solution is `x in (-oo,(3-sqrt3)/2] uu [(3+sqrt3)/2,+oo)`

Explanation:

We need the roots of the equation

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

The discriminant is

`Delta=b^2-4ac=6^2-4*2*3=36-24=12`

As, `Delta>0`, there are 2 real roots

`x_1=(6-sqrt12)/(2*2)=(6-2sqrt3)/(4)=(3-sqrt3)/(2)`

`x_2=(6+sqrt12)/(2*2)=(6+sqrt3)/(4)=(3+sqrt3)/(2)`

Let our inequality be

`f(x)=(x-x_1)(x-x_2)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `x_1` `color(white)(aaaa)` `x_2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x-x_1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-x_2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)>=0` when `x in (-oo,(3-sqrt3)/2] uu [(3+sqrt3)/2,+oo)`
graph{2x^2-6x+3 [-4.93, 4.934, -2.465, 2.465]}