if the roots of the equation a minus b x square + b minus C X + c minus A is equal to zero of equal prove that to a equal to b + c
Answers
Answer:
Step-by-step explanation:
We know that, If the quadratic equation ax²+bx+c=0 whose roots are equal then it's deteminant is equal to zero.
(a-b)x² + (b-c)x + (c-a) = 0
(b-c)² - 4 (a-b)(c-a) = 0
b²+c² - 2bc - 4(ac - a² -bc +ba) = 0
b²+ c² -2bc + 4a²+4bc-4ac-4ab=0
b²+ c² +2bc+ 4a²-4ac-4ab = 0
b²+ c² + 2bc + (-2a)² + 2(-2a)(c)+2(-2a)(b) = 0
(b + c - 2a)² = 0
b + c - 2a = 0
2a = b + c
Hence proved.
Answer:
Step-by-step explanation:
Solution :-
(a - b)x² + (b - c)x + (c - a) = 0
The roots are equal, then D = 0
Comparing Eq by ax² + bx + c = 0
a = (a - b), b = (b - c), c = (c - a)
D = b² - 4ac
= (b - c)² - 4 × (a - b) (c - a)
Here, D = 0
⇒ (b - c)² - 4(a - b) (c - a) = 0
⇒ b² + c² - 2bc - 4(ac - a² - bc + ab) = 0
⇒ b² + c² - 2bc - 4ac + 4a² + 4bc - 4ab = 0
⇒ 4a² + b² + c² + 2bc - 4ab - 4ac = 0
⇒ (- 2a + b + c)² = 0 [a² + b² + c² + 2ab + 2bc + 2ca = (a + b + c)²]
⇒ b + c = 2a
Hence, Proved.