How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given #3/4x^2-1/3x-1=0#?

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Answer 1 · 2018-05-09T15:51:31

#Delta=1/9+3=28/9>0=> "two distinct real roots"#

#x=2/9+4/9sqrt7#
or
#x=2/9-4/9sqrt7#

for a quadratic equation

#ax^2+bx+c=0#

the discriminant is given by

#Delta=b^2-4ac---(1)#

#Delta>0=>#two distinct real roots

#Delta=0=>#one root (ie two equal roots)

#Delta<0=># two distinct complex roots( conjugate pairs if #a,b,cinRR#)

we have

#3/4x^2-1/3x-1=0#

multiply the eqn by#12#

#a=3/4,b=-1/3, c=-1#

#(1)rarrDelta=(-1/3)^3-4(3/4)(-1)#

#Delta=1/9+3=28/9>0#

#:. two distinct real roots

#quadratic eqn

#x=(-b+-sqrtDelta)/(2a)#

#x=(1/3+-sqrt(28/9))/(3/2)#

#x=(1/3+-(2sqrt7)/3)/(3/2)#

#x=2/9+-4/9sqrt7#