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If the Roots of the Equation (B - C)X2 + (C - A)X + (A - B) = 0 Are Equal, Then Prove that 2b = a + C. - CBSE Class 10 - Mathematics

Question

If the roots of the equation (b - c)x2 + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.

Solution

The given quadric equation is(b - c)x2 + (c - a)x + (a - b) = 0, and roots are real

Then prove that 2b = a + c

Here,

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

As we know that D = b2 - 4ac

Putting the value of a = (b - c), b = (c - a) and c = (a - b)

D = b2 - 4ac

= (c - a)2 - 4 x (b - c) x (c - a)

= c2 - 2ca + a2 - 4 (ab - b2 - ca + bc)

= c2 - 2ca + a2 - 4ab + 4b2 + 4ca - 4bc

= c2 + 2ca + a2 - 4ab + 4b2 - 4bc

= a2 + 4b2 + c2 + 2ca - 4ab - 4bc

As we know that (a2 + 4b2 + c2 + 2ca - 4ab - 4bc) = (a + c - 2b)2

D = (a + c - 2b)2

The given equation will have real roots, if D = 0

(a + c - 2b)2 = 0

Square root both side we get

sqrt((a + c - 2b)^2)=0

a + c - 2b = 0

a + c = 2b

Hence 2b = a + c.

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Solution If the Roots of the Equation (B - C)X2 + (C - A)X + (A - B) = 0 Are Equal, Then Prove that 2b = a + C. Concept: Nature of Roots.
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