Find the possible values of k.

Question

The equation x2 + (k+2)x + 2k = 0 has two distinct roots. Find the possible values of k.

Answer 1

To have 2 distinct roots, the discriminant must not be zero.

 

x² + (k + 2)x + 2k = 0

 

D = (k + 2)² - 4(1)(2k) = (k² + 4k + 4) - 8k = k² - 4k + 4 = (k - 2)² ≠ 0

 

Any real k ≠ 2 will ensure 2 distinct roots.

Answer 2

Two distinct roots means the discriminant is greater than zero.

 

The discriminant is

  b^2 - 4ac > 0  where A=1 B=(k+2)  C = 2k

 

So 

 

0 < discriminant =

 

 (k+2)^2  - 4(1)(2k) =

 

 (k+2)^2 - 8k

 

 k^2 + 4k + 4 - 8k 

 

k^2 - 4k + 4

 

 

Then   (k   - 2)(k-2)>0

 

 For k=2, the inequality is false, as (2-2)(2-2)>0

                                                        0 * 0 > 0

                                                             0 > 0 <--- FALSE

 

If k<2, then k-2 < 0

                (k-2)(k-2) > 0  <--- reverses inequality when multiplying by negative number

 So for k<2, the inequality is true

 

if k>2, then k-2>0

 (k-2)(k-2)>0

So for k>2 the inequality is true

 

So the quadratic equation has two distinct roots for k not equal to 2.