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

1 Answer
Jun 28, 2017

Answer:

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]}