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4x^(3)+72x^(2)=-324 x

$4 x^{3}+72 x^{2}=-324 x$

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Find derivatives of using multiple formulae

Full solution

Q.

4 x^{3}+72 x^{2}=-324 x

Bring terms together: First, we need to bring all terms to one side of the equation to set it equal to $0$ . $4x^3 + 72x^2 + 324x = 0$
Factor out common factor: Next, we can factor out the greatest common factor, which is $4x$ in this case. $\newline$ $4x(x^2 + 18x + 81) = 0$
Factor quadratic equation: Now, we need to factor the quadratic equation if possible. $x^2 + 18x + 81$ is a perfect square trinomial, which factors into $(x + 9)^2$ . $4x(x + 9)^2 = 0$
Set factors equal to zero: We can now set each factor equal to zero to find the roots of the equation. $\newline$ First, set $4x$ to zero: $\newline$ $4x = 0$ $\newline$ $x = 0$
Find roots: Then, set $(x + 9)^2$ to zero: $\newline$ $(x + 9)^2 = 0$ $\newline$ $x + 9 = 0$ $\newline$ $x = -9$
Identify repeated root: Since $(x + 9)^2$ is a square, it gives us the same root twice, so $x = -9$ is a repeated root. $\newline$ Therefore, the roots of the equation are $x = 0$ and $x = -9$ (with a multiplicity of $2$ ).

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