If $a x^{2} + b x + c = 0, b x^{2} + c x + a = 0$ have common root $1$ and $a, b, c$ are zero real no. then find the value of $\frac{a^{3} + b^{3} + c^{3}}{3 a b c}$ .

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Solution

Given, $1$ is a common root of $a x^{2} + b x + c = 0, b x^{2} + c x + a = 0$ , then we have from both equation (by satisfying $x = 1$ in the equations),

$a + b + c = 0$

or, $a + b = - c$ .......(1).

Now cubing both sides we get,

$(a + b)^{3} = - c^{3}$

or, $a^{3} + b^{3} + 3 a b (a + b) = - c^{3}$

or, $a^{3} + b^{3} - 3 a b c = - c^{3} = 0$ [ Using (1)]

or, $a^{3} + b^{3} + c^{3} = 3 a b c$

or, $\frac{a^{3} + b^{3} + c^{3}}{3 a b c} = 1$ .

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If ax2+bx+c=0,bx2+cx+a=0 have common root 1 and a,b,c are zero real no. then find the value of a3+b3+c33abc.

Given, 1 is a common root of ax2+bx+c=0,bx2+cx+a=0 , then we have from both equation (by satisfying x=1 in the equations),a+b+c=0or, a+b=−c.......(1).Now cubing both sides we get,(a+b)3=−c3or, a3+b3+3ab(a+b)=−c3or, a3+b3−3abc=−c3=0 [ Using (1)]or, a3+b3+c3=3abcor, a3+b3+c33abc=1.

If $a x^{2} + b x + c = 0, b x^{2} + c x + a = 0$ have common root $1$ and $a, b, c$ are zero real no. then find the value of $\frac{a^{3} + b^{3} + c^{3}}{3 a b c}$ .