Find roots of polynomial $$ p(x) = 32x-48x^2 $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= \frac{ 2 }{ 3 } \end{aligned} $$

Step 1:

Write polynomial in descending order

$$ \begin{aligned} 32x-48x^2 & = 0\\[1 em] -48x^2+32x & = 0 \end{aligned} $$

Step 2:

Factor out $ \color{blue}{ -16x }$ from $ -48x^2+32x $ and solve two separate equations:

$$ \begin{aligned} -48x^2+32x & = 0\\[1 em] \color{blue}{ -16x }\cdot ( 3x-2 ) & = 0 \\[1 em] \color{blue}{ -16x = 0} ~~ \text{or} ~~ 3x-2 & = 0 \end{aligned} $$

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

Step 3:

To find the second zero, solve equation $ 3x-2 = 0 $

$$ \begin{aligned} 3x-2 & = 0 \\[1 em] 3 \cdot x & = 2 \\[1 em] x & = \frac{ 2 }{ 3 } \end{aligned} $$

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