How do you use the discriminant to determine the nature of the solutions given `2x^2+4x+1=0`?

`color(orange)"Reminder"`

Rational numbers `QQ` are numbers which can be written in the form `a/b` where a and b are integers `ZZ`

Numbers such as `5/2,-2/3,6=6/1" are rational"` otherwise they are irrational, `4.bar6,sqrt5,pi" etc"` are irrational.

The discriminant `Delta=b^2-4ac` informs us about the nature of the roots.

`•b^2-4ac>0rArrcolor(blue)"roots are real and irrational"`

`•b^2-4ac>0" and a square"rArrcolor(blue)" roots are real and rational"`

`•b^2-4ac=0rArrcolor(blue)" roots are real/rational and equal"`

`•b^2-4ac<0rArrcolor(blue)"roots are not real"`

For `2x^2+4x+1=0 rArra=2,b=4,c=1`

`rArrb^2-4ac=4^2-(4xx2xx1)=16-8=8>0`

Since discriminant > 0 , roots are real and irrational.
`color(red)"-----------------------------------------------------------"`

As a check for you,let's solve the equation using the `color(magenta)" quadratic formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(x=(-b±sqrt(b^2-4ac))/(2a))color(white)(a/a)|)))`
Using the values of a , b and c from above.

`rArrx=(-4±sqrt8)/4`

The roots are x = -0.293 and x = -1.707 (to 3 decimal places)

Thus roots are real and irrational as predicted.