Question

a, b, c are co-prime a not equal to 1 such that 2b = a + c. If ax^2-2bx+c and 2x^3-5x^2 + kx + 4 has one integral root common, then find the value of k.

Answer 1

Step-by-step explanation:

Given A, b, c are co-prime a not equal to 1 such that 2 b = a + c. If ax^2-2 bx+c and 2 x^3-5 x^2 + k x + 4 has one integral root common, then find the value of k.

• According to question a is not equal to 1 and abc are co-prime. Both equations have one common root which is 1.
• 2 b = a + c
• Also ax^2 – 2 bx + c = 0
• So ax^2 – (a + c)x + c = 0
• So ax^2 – ax – c x + c = 0
• So ax(x – 1) – c(x – 1) = 0
• Or (x – 1) (ax – c) = 0
• So x = 1, x = c/a (here c,a are co-primes)
• Therefore put x = 1
• Now 2x^3 – 5x^2 + kx + 4 = 0
•     2(1)^3 – 5(1)^2 + k(1) + 4 = 0
•      2 – 5 + k + 4 = 0
•     1 + k = 0
• Or k = - 1

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Answer 2

2 b = a + c

ax^2 – 2 bx + c = 0

So ax^2 – (a + c)x + c = 0

So ax^2 – ax – c x + c = 0

So ax(x – 1) – c(x – 1) = 0

Or (x – 1) (ax – c) = 0

So x = 1, x = c/a (here c,a are co-primes)

Therefore put x = 1

Now 2x^3 – 5x^2 + kx + 4 = 0

    2(1)^3 – 5(1)^2 + k(1) + 4 = 0

     2 – 5 + k + 4 = 0

    1 + k = 0

Or k = - 1