Question

If roots of b minus c into x square + c minus a into x + 3 = 0 are equal then prove that a b c are in ap

Answer 1

$\huge{\boxed{\mathtt{\red{ANSWER}}}}$

If roots of a quadratic equation are equal, then the discriminant of the quadratic equation is 0.

Than ,

D = b²-4ac

Here, a =(b-c) , b = (c-a) and c = (a-b)

so,

D = (c−a)²−4(b−c)(a−b)=0

c²+a²−2ac−4(ab−b²−ac+bc)=0

c²+a²−2ac−4ab+4b²+4ac−4bc = 0

c²+a²+2ac+4b²−4ab−4bc = 0

(c+a)²+4b²−4b(a+c) = 0

(c+a)²+(2b)²−2⋅(c+a)⋅(2b) = 0

[(c+a)−(2b)]² = 0

⇒ c+a−2b = 0

a + c = 2b

so, a , b & C are in AP ...

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