If f(x) = 2x^3 + mx^2 - 13x + n and 2, 3 are | vedclass

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(C) Given $f(x) = 2x^3 + mx^2 - 13x + n = 0$. Since $2$ and $3$ are roots,$f(2) = 0$ and $f(3) = 0$.For $x = 2$: $2(2)^3 + m(2)^2 - 13(2) + n = 0 \imp

Vedclass · Accepted Answer

$130$

Vedclass · Answer

(C) Given $f(x) = 2x^3 + mx^2 - 13x + n = 0$. Since $2$ and $3$ are roots,$f(2) = 0$ and $f(3) = 0$.For $x = 2$: $2(2)^3 + m(2)^2 - 13(2) + n = 0 \implies 16 + 4m - 26 + n = 0 \implies 4m + n = 10$.For $x = 3$: $2(3)^3 + m(3)^2 - 13(3) + n = 0 \implies 54 + 9m - 39 + n = 0 \implies 9m + n = -15$.Subtracting the first equation from the second: $(9m + n) - (4m + n) = -15 - 10 \implies 5m = -25 \implies m = -5$.Substituting $m = -5$ into $4m + n = 10$: $4(-5) + n = 10 \implies -20 + n = 10 \implies n = 30$.We need to find $4m + 5n$: $4(-5) + 5(30) = -20 + 150 = 130$.

If $f(x) = 2x^3 + mx^2 - 13x + n$ and $2, 3$ are the roots of the equation $f(x) = 0$,then the value of $4m + 5n$ is:

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