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The coefficient of $n^2$ of the expanded form of $(5n^2 - 4n - 3)^2$ is $-14$ .

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Find values of derivatives using limits

Full solution

Q. The coefficient of

n^2

of the expanded form of

(5n^2 - 4n - 3)^2

is

-14

.

Identify Terms for Squaring: Identify the terms in the binomial that will produce $n^2$ terms when squared. $\newline$ Calculation: $(5n^2 - 4n - 3)^2 = (5n^2)^2 + 2(5n^2)(-4n) + 2(5n^2)(-3) + (-4n)^2 + 2(-4n)(-3) + (-3)^2$
Calculate Coefficients: Calculate the coefficient of $n^2$ from each relevant term. $\newline$ Calculation: From $(5n^2)^2$ , coefficient is $25$ . $\newline$ From $(-4n)^2$ , coefficient is $16$ . $\newline$ From $2(5n^2)(-4n)$ , no $n^2$ term. $\newline$ From $2(5n^2)(-3)$ , no $n^2$ term. $\newline$ From $2(-4n)(-3)$ , no $n^2$ term. $\newline$ From $(5n^2)^2$ $1$ , no $n^2$ term. $\newline$ Total coefficient of $n^2$ = $25$ + $16$ = $41$ .

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