Question

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

Please don't Answer BullSh*t!​

Answer 1

Answer:

2b = a + c

Step-by-step explanation:

It is given that roots are equal.

This happens only when the value of Discriminant is equal to zero.

We know that:

Quadratic equation is of the form ax² + bx + c = 0

Here,

• a = (b - c)
• b = (c - a)
• c = (a - b)

D = 0

⇒ b² - 4ac = 0

⇒ (c - a)² - 4(b - c) (a - b) = 0

⇒ (c - a)² - 4 [ab - b² - ac + bc] = 0

⇒ c² + a² - 2ac - 4ab + 4b² + 4ac - 4bc = 0

⇒ c² + a² + 4ac - 2ac - 4ab + 4b² - 4bc = 0

⇒ (c + a)² - 4ab + 4b² - 4bc = 0

⇒ (c + a)² + 4b² - 4ab - 4bc = 0

⇒ (c + a)² + (2b)² - 2 (2b (c + a)) = 0

This is of the form (a² + b² - 2ab) which is equal to (a - b)².

So,

→ (c + a - 2b)² = 0

⇒ (c + a - 2b) = 0

⇒ c + a = 2b

⇒ 2b = a + c

★ HENCE PROVED ★

Answer 2

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