F.4 Math quadratic equation40

Find the discriminant of (*)
△ = (2k)^2 - 4(k^2)(1)
= 4k^2 - 4k^2
= 0
Find the value of αβ and α+β in terms of k
Since △ = 0 ,we have αβ = α+β
k^2x^2 + 2kx+1 = 0
(kx + 1)^2 = 0
kx + 1 = 0
x = - 1 / k
x = αβ = α+β = - 1 / k
Find a quadractic equation in x whose roots are α and β
Ifβ =2, find the value of k and α
αβ = α+β
2α = α+ 2
α = 2
- 1 / k = α+β or αβ = 2 + 2 or 2*2 = 4
- 1 = 4k
k = - 1 / 4





2009-10-04 00:22:40 補充：
Sorry!I missing the question :{Find a quadractic equation in x whose roots are α and β}

The equation is x^2 - (α+β)x + αβ = 0 , since αβ = α+β = - 1 / k ,

the equation becomes 『x^2 + (1/k)x - 1/k = 0』

2009-10-04 00:23:15 補充：
Ifβ =2, find the value of k and α?

2^2 + (1/k)(2) - 1/k = 0

1/k = - 4

『k = - 1/4』

[α(2) = α+ 2] ------------------------------(= - 1 / k = - 1 / (-1/4) = 4 )

『α = 2』------------------------------------In fact ,(2*2 = 2+2 = 4)

2009-10-04 00:25:04 補充：
the equation becomes 『x^2 + (1/k)x - 1/k = 0』

i.e. 『 kx^2 + x - 1 = 0 』

2009-10-04 00:45:15 補充：
Sub k = -1/4 to 『 kx^2 + x - 1 = 0 』

(-1/4)x^2 + x - 1 = 0

x^2 - 4x + 4 = 0

(x - 2)^2 = 0