{x}^(2) + (k - 2)x - 2k = 0 Prove that the roots of the equation x2 + (k − 2)x – 2k = 0 are real and distinct for all real values of k. ​

Question

`{x}^(2) + (k - 2)x - 2k = 0`

Prove that the roots of the equation x2 + (k − 2)x – 2k = 0 are real and distinct for all real values of k. ​

Answer 1

Explanation:

Use the discriminate of the quadratic equation.

a = 1

b = k - 2

c = - 2k

Discriminate (D) = sqrt(b^2 - 4*a*c)

D = sqrt( (k - 2)^2 - 4(1)(-2k) )

D = sqrt( k^2 -4k + 4 + 8k)

D = sqrt(k^2 +4k + 4)

D = sqrt(k + 2)^2

D = (k+2)

The domain of k can be any real number -- nothing is excluded.