Proving-polynomials (1 Viewer)

[shaon0]

Prove that the roots of the equation kx(x+4)=(x-1)(x+2) are always real.
I cannot prove this and need help. All help will be appreciated
I keep getting unreal values in this.

 

[Iruka]

Work out the discriminant, which is a quadratic in k. Then show that this quadratic is positive definite.

 

[shaon0]

Work out the discriminant, which is a quadratic in k. Then show that this quadratic is positive definite.

lol...i still keep getting unreal values for the discriminant.

 

[Iruka]

OK, so kx(x+4)=(x-1)(x+2)

Then (k-1)x^2 + (4k-1)x +2=0
so the discriminant is (4k-1)^2 - 8(k-1)= 16k^2 -16k +9.

If we complete the square, we get

16k^2 -16k +9 = (4k - 2)^2 +5 > 0 for all k.

 

[tommykins]

回复: Proving-polynomials

Prove that the roots of the equation kx(x+4)=(x-1)(x+2) are always real.
I cannot prove this and need help. All help will be appreciated
I keep getting unreal values in this.

kx^2 + 4kx = x^2 + x - 2

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

Discriminant

(4k-1)^2 - 8.(k-1)

= 16k^2 - 8k + 1 - 8k + 8
= 16k^2 - 16k + 9
= (4k-2)^2 + 5

(4k-2)^2 + 5 is always > 0 as (4k-2)^2 > 0 and 5 > 0

Thus polynomial always has real roots.

 

[shaon0]

Re: 回复: Proving-polynomials

thanks guys for your help. really appreciate it.
Thanks