find whether the following equations have real roots if real roots exist find them.1/2x-3+1/x-5=1x not equal to 3/2 ,5
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Answer:
Main concept used: For real roots of quadratic equationax
2
+bx+c=0,b
2
−4ac>0
(iv)
(2x−3)
1
+
(x−5)
1
=1,x
=
2
3
,5
(2x−3)
1
+
(x−5)
1
=1,x
=
2
3
,5
⇒
(2x−3)(x−5)
(x−5)+(2x−3)
=1
⇒x−5+2x−3=2x
2
−3x−10x+15
⇒2x
2
−13x+15=3x−8
⇒2x
2
−13x+15−3x+8=0
⇒2x
2
−16x+23=0
Now, D=b
2
−4ac
=(−16)
2
−4(2)(23)(a=2,b=−16,c=23)
D=256−184=72>0
⇒
D
=
72
⇒
D
=6
2
Now, x=
2a
−b±
D
=
2×2
16±6
2
=
4
16
±
4
6
2
⇒x
1
=4+
2
3
2
and x
2
=4−
2
3
2
Hence, the root of the given quadratic equation are (4+
2
3
2
) and (4−
2
3
)
2
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