Find roots of polynomial $$ p(x) = -4x^3+9x^2 $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{ 9 }{ 4 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ -x^2 }$ from $ -4x^3+9x^2 $ and solve two separate equations:

$$ \begin{aligned} -4x^3+9x^2 & = 0\\[1 em] \color{blue}{ -x^2 }\cdot ( 4x-9 ) & = 0 \\[1 em] \color{blue}{ -x^2 = 0} ~~ \text{or} ~~ 4x-9 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 2:

To find the last zero, solve equation $ 4x-9 = 0 $

$$ \begin{aligned} 4x-9 & = 0 \\[1 em] 4 \cdot x & = 9 \\[1 em] x & = \frac{ 9 }{ 4 } \end{aligned} $$

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