Quadratic equations~provement

用戶36732

2009-08-28 7:54 pm

(1)
(a) It is given that k is a non-zero constant. Express the discriminant of the quadratic equation k(x^2) + (k+3)x + 3 = 0 in terms of k.
(b) Prove that the equation has real roots.

(2)
It is given that [(k^2)+1]x^2 = 3x-4 is an equation in x. Prove that the equation has no real roots for all real numbers k.

回答 (1)

螞蟻雄兵

2009-08-28 8:08 pm

✔ 最佳答案

(1)
(a)It is given that k is a non-zero constant. Express the discriminant of thequadratic
equationk(x^2) + (k+3)x + 3 = 0 in terms of k.
D=(k+3)^2－4*k*3
=k^2+6k+9－12k
=k^2－6k+9

(b)Prove that the equation has real roots.
D=k^2－6k+9
=(k－3)^2>=0
So has real roots

(2)Itis given that [(k^2)+1]x^2 = 3x－4 is an equation in x. Provethat the equation has no
realroots for all real numbers k.
(k^2+1)x^2=3x－4
(k^2+1)x^2－3x+4=0
D=(－3)^2－4*(k^2+1)*4
=9－16k^2-16
=－16k^2－7
Since k^2>=0 for all real numbers k.
－k^2<=0
－16k^2<=0
－16k^2－7<=－7<0
SoD<0
So has no real roots for all real numbers k.

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