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

1 Answer
Jul 20, 2016

Answer:

roots are real and irrational.

Explanation:

#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.