Home > Algebra > Solutions Using the Discriminant

How do you describe the nature of the roots of the equation #7x^2=4x+1#?

Answer 1

Nathan Axel

Roots are irrational. In the given equation these are
#-2/7-sqrt11/7# and
#-2/7+sqrt11/7#

Nature of roots of an equation #ax^2+bx+c=0#/ are determined by its discriminant, which is #b^2-4ac#.

Assuming that the coefficients #a#, #b# and #c# are rational,

if discriminant is zero, there is one root and we also say roots coincide.

if discriminant is negative, roots are complex,

if discriminant is positive and square of a rational number, roots are rational

and if discriminant is positive but not a square of a rational number, roots are irrational.

As the equation #7x^2=4x+1# #hArr7x^2-4x-1=0#, the discriminant is #(-4)^2-4×7×(-1)=16+28=44#, which is positive but not a square of a rational number. Hence roots are irrational

In fact, according to quadratic formula roots are

#((-4)+-sqrt44)/(2×7)# or

#-4/14+-2sqrt11/14# or

#-2/7-sqrt11/7# and #-2/7+sqrt11/7#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.