given 3 and m are the roots of the equation(2x-1) (x+4)=k(x-1)+3,where k is a constant.find value of m and k?

Question

please show your working and explain it in details.thank you.

Answer 1

Multiply it out - it'll be quadratic.

2x^2 +7x - 4 = kx + 3-k

2x^2 + (7 - k)x + (k - 7) = 0

If the roots are 3 and m, then the factors must be (x - 3)(x - m), so the equation would be

x^2 - (m + 3)x + 3m = 0

These two equations must be identical. The coefficients must match up exactly, so we multiply the second one by 2:

2x^2 - 2(m + 3)x + 6m = 0

Therefore,

-2(m + 3) = 7 - k

and

6m = k - 7

Two equations, two unknowns - this can be solved. You should work out the solution to verify it, but it turns out m = 3/2 and k = 16.

Answer 2

1.Substitute 3 for x in the equation, to get
5*7=2k+3 and so k=16.
2.Substitute m for x and 16 for k in the equation, to get
(2m-1)(m+4)=16(m-1)+3 or, 2m^2+7m-4=16m-13
2m^2-9m+9=0 which is same as 2(m-3)(m-1.5)=0, and so
m=3 or m=1.5.