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Q.

Find whether the following equation have real roots. If yes, find them.


4 x 2 − 1 5 x+ 1 32 =0


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a

  the equation 4 x 2 − 1 5 x+ 1 32 =0 does have real roots and x=4 or x=-4.

b

  the equation 4 x 2 − 1 5 x+ 1 32 =0 does have real roots and x=32 or x=5.

c

  the equation 4 x 2 − 1 5 x+ 1 32 =0 does have real roots and x= 2 or x=5.

d

  the equation 4 x 2 − 1 5 x+ 1 32 =0 does not have real roots. 

answer is D.

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Detailed Solution

Given equation is 4 x 2 − 1 5 x+ 1 32 =0.
We know that a polynomial's discriminant is a number that depends on the coefficients and defines several features of the roots.
For a quadratic equation,
(i) If the discriminant is positive, two separate roots exist.
(ii) Both roots are equivalent if the discriminant is zero.
(iii) There are no real roots if the discriminant is negative. Instead, there are two different (non-real) complex roots.
In quadratic equation, a x 2 +bx+c=0(a≠0) Discriminant, D= b 2 −4ac Root of quadratic equation, x= −b± D 2a .
The discriminant of the above equation is
 Discriminant,
D = −1 5 2 −4(4) 1 32 = 1 25 − 1 2 = 2−25 50 = −23 50 Here, we can observe that D<0.
Hence, the equation 4 x 2 − 1 5 x+ 1 32 =0 does not have real roots.
The correct option is (4).
 
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Find whether the following equation have real roots. If yes, find them. 4 x 2 − 1 5 x+ 1 32 =0