2. If a and b are distinct real numbers, show that the quadratic equation
2(a + b2) x2 + 2(a + b)x + 1 = 0 has no real roots.
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If a and b are distinct real numbers, show that the quadratic equation 2(a
2
+b
2
)x
2
+2
(a+b)x+1=0 has no real roots.
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Answer
2(a
2
+b
2
)x
2
+2(a+b)x+1=0
Compare given equation with the general form of quadratic equation, which ax
2
+bx+c=0
a=2(a
2
+b
2
),b=2(a+b),c=1
Discriminant:
D=b
2
−4ac
=[2(a+b)]
2
−4.2(a
2
+b
2
).1
=4a
2
+4b
2
+8ab−8a
2
−8b
2
=−4a
2
−4b
2
+8ab
=−4(a
2
+b
2
−2ab)
=−4(a−b)
2
<0
Hence the equation has no real roots.
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