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

##### 1 Answer

#### 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.