1. ## Need help for Proving Quadratic Equations

Hello people,

How do i prove these?

1)
If a, b and c are real, show that the roots of the following equations are real:

(i) (b+c)x^2 - (a + b + c)x + a = 0
(ii) (a + b)(x^2 + 5x + 1) = c(x^2-1)

2)
If roots of the equation (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 are equal, prove that a = b = c.

3) If a = b = c, prove that the roots of the equation:

(x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 are equal.

2. 1)
i) The discriminant is

$\displaystyle \Delta=(a+b+c)^2-4a(b+c)=a^2+b^2+c^2-2ab-2ac+2bc=(a-b-c)^2\geq 0$, so the roots are real

ii) The equation is $\displaystyle (a+b-c)x^2+5(a+b)x+a+b+c=0$

$\displaystyle \Delta=25(a+b)^2-4(a+b+c)(a+b-c)=25(a+b)^2-4(a+b)^2+4c^2=21(a+b)^2+4c^2\geq 0$

2) The equation is

$\displaystyle 3x^2-2(a+b+c)x+ab+ac+bc=0$

If the roots are equal then the discriminant is 0.

$\displaystyle \Delta=4(a+b+c)^2-12(ab+ac+bc)=4(a^2+b^2+c^2-ab-ac-bc)=$

$\displaystyle =2[(a-b)^2+(a-c)^2+(b-c)^2]$

$\displaystyle \Delta=0\Rightarrow a-b=0, \ a-c=0, \ b-c=0\Rightarrow a=b=c$

3) $\displaystyle a=b=c\Rightarrow3(x-a)^2=0\Rightarrow x_1=x_2=a$

3. Cool