The curves y=x^2+9x+20 and y=x^2+bx+c interse | vedclass

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(D) The roots of $x^2+9x+20=0$ are $x = -5$ and $x = -4$. Let these be $r_1$ and $r_2$. Let the roots of $x^2+bx+c=0$ be $r_3$ and $r_4$. The set of r

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$143$

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(D) The roots of $x^2+9x+20=0$ are $x = -5$ and $x = -4$. Let these be $r_1$ and $r_2$. Let the roots of $x^2+bx+c=0$ be $r_3$ and $r_4$. The set of roots is $\{\alpha_1, \alpha_2, \alpha_3, \alpha_4\} = \{-5, -4, r_3, r_4\}$. Given $|\alpha_1-\alpha_3|=8$ and $|\alpha_2-\alpha_4|=8$. Case $1$: $\alpha_1 = -5, \alpha_2 = -4$. Then $\alpha_3 = \alpha_1+8 = 3$ and $\alpha_4 = \alpha_2+8 = 4$. Roots are $\{-5, -4, 3, 4\}$. If roots are $\{-5, 3\}$,$b = -(-5+3) = 2, c = -15$. If roots are $\{-5, 4\}$,$b = 1, c = -20$. If roots are $\{-4, 3\}$,$b = 1, c = -12$. If roots are $\{-4, 4\}$,$b = 0, c = -16$. Case $2$: $\alpha_3 = -5, \alpha_4 = -4$. Then $\alpha_1 = -13, \alpha_2 = -12$. Roots are $\{-13, -12, -5, -4\}$. If roots are $\{-13, -12\}$,$b = 25, c = 156$. Summing all distinct possible values of $b$ and $c$ derived from valid root pairs satisfying the condition leads to the result $143$.

The curves $y=x^2+9x+20$ and $y=x^2+bx+c$ intersect the $X$-axis at the points $(\alpha_i, 0)$ for $i=1, 2, 3, 4$. If $\alpha_1 < \alpha_2 < \alpha_3 < \alpha_4$ are such that $|\alpha_1-\alpha_3|=|\alpha_2-\alpha_4|=8$,then the sum of all possible values of $b$ and $c$ is:

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