let a, b and c are unequal real positive numbers such that 2b = a + c, then roots of ax^2 + 2bx + c = 0 are


(i) real and equal (ii) Real and distinct (iii) imaginary (iv) nothing definite can be said


Plz explain

3 years ago

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Answers : (3)

                    

Dear student,


2b=a+c


Sum of the roots=-2b/a


Product=c/a


This shows the roots are real and distinct...








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3 years ago
                    

for any quadratic eq  ax2+bx+c roots are given by


 X = {-b+_ D)/2a


D =discriminant = (b2-4ac)1/2


our eq is ax2+2bx+c so


D = {(2b)2-4ac)1/2 ........1


2b = a+c  (given)


so , (2b)2 = (a+c)2 ...........2


putting  eq2 in eq1


D = { (a+c)2 - 4ac}1/2  =  { a2+c2-2ac}1/2 ={(a-c)2}1/2 =a-c


now roots are


 X = (-2b +_ D}/2a = {-2b + (a-c)}/2a     or        {-2b-(a-c)}/2a                                   


                           = -c/a                      or        -1                                  (on putting 2b =a+c )


so the roots are unequal & real


option (ii) is correct

3 years ago
                    

the equation is ax^2+2bx+c=0


the discriminant is= B^2-4AC


A=a,B=2b,C=c


discriminant= 4b^2-4ac


but 2b=a+c, therefore (2b)^2=(a+c)^2


= (a+c)^2-4ac


=a^2+c^2+2ac-4ac


=a^2+c^2-2ac


=(a-c)^2, which is always greater than zero


therefore roots are real and distinct

3 years ago

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