When k takes what value, the equation x * x + kx-3 = 0 and the equation x * x + x-3k = 0 have a common root? Find out the common root "X * x" is the square of X

When k takes what value, the equation x * x + kx-3 = 0 and the equation x * x + x-3k = 0 have a common root? Find out the common root "X * x" is the square of X

Let the common root be a
Then:
a*a+ka-3=0
a*a+a-3k=0
The result of subtracting the two formulas is as follows:
(k-1)a+(3k-3)=0
Namely:
(k-1)(a+3)=0
When k = 1
The two equations are reduced to a * a + A-3 = 0
Two common roots are obtained from the root formula
A1 = (- 1 + radical 13) / 2
A2 = (- 1-radical 13) / 2
When k is not equal to 1
Common root:
a=-3

When k takes what value (K ≠ 1), the equation x ^ 2 + KX + 3 = 0 and the equation x ^ 2 + X + 3K = 0 have a common root
RT

By subtracting the two equations: (k-1) x + 3-3k = 0
Because K1, x = 3
This is the common root

When k is what value, the equation X2 - (K + 2) x + 12 = 0 and the equation 2x2 - (3K + 1) x + 30 = 0 have a common root? Find the common root

Let t be the common solution. According to the meaning of the question, T2 - (K + 2) t + 12 = 0, 2t2 - (3K + 1) t + 30 = 0, ②, ① × 2 - ② get (K-3) t = 6, when K-3 ≠ 0, t = 6K − 3, substitute t = 6K − 3 into ①, get (6K − 3) 2 - (K + 2) · 6K − 3 + 12 = 0, sort out k2-11k + 30 = 0, get K1 = 5, K2 = 6, when k = 5, t = 3; when k = 6, t = 2, that is, when k is 5, equation X2 - (K + 2) x + 12 = 0 and equation 2x2 - (3K + 1) X +When k is 6, the equation X2 - (K + 2) x + 12 = 0 and the equation 2x2 - (3K + 1) x + 30 = 0 have a common root 2