Question

Find the value of k,if 3x-4 is a factor of expression (3k+2)x^3- (k-1)​

Answer 1

First divide by k^2 so the coefficient of x^2 is 1

f(x) = x^2 +2(k+1)x/k^2 +4/k^2

To complete the square divide the co efficient of x by 2 to get (k+1)/k^2

Then complete the square f(x) = [ x+(k+1)/k^2]^2

Expand to get

f(x) = x^2 + 2(k+1)x/k^2 +(k+1)^2/k^4

Now the constant (k+1)^2/k^4 = 4/k^2 ,the constant in f(x) ,cancel k^2 and cross multiply by k^2

Then (k+1)^2 =4k^2

k^2+2k+1 = 4k^2

3k^2 -2k-1 =0

(3k+1)(k-1) =0

k={ -1/3 , 1} is the solution set for k

Eg k=1 then f(x)= x^2 +4x+4 =(x+2)^2

k=(-1/3) then f(x) =(1/9)x^2+ 12/9x+ +4 = (x/3 +2)^2 =[(1/3)(x+2/3)]^2

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