To simplify the expression \( 36x - 216x^2 - 36 \), we start by factoring the expression.
- First, we can factor out a common term from each of the terms in the expression. The greatest common factor here is \( 36 \):
\[ 36x - 216x^2 - 36 = 36(x - 6x^2 - 1) \]
- Now let's rewrite the expression inside the parentheses:
\[ x - 6x^2 - 1 = -6x^2 + x - 1 \]
To reorder it, we get:
\[ -6x^2 + x - 1 \]
- Now, factor out a negative:
\[ -(6x^2 - x + 1) \]
This quadratic doesn't factor easily, but we don't need to factor it completely to express the original expression in simplest form.
Putting it all together, we go back to our factored expression:
\[ 36(-6x^2 + x - 1) \]
However, now this seems more complicated. Let's go back and simplify the original expression directly.
We actually want to see if we can factor directly from the beginning:
The expression was:
\[ 36x - 216x^2 - 36 \]
We can factor each term over the common factor:
\[ = 36(x - 6x^2 - 1) \]
To confirm we can find zeroes or intercepts by solving \( 6x^2 - x - 1 \). This quadratic has two zeroes which we can use:
However, the options seem to favour a simpler path. So let's examine the expression in general forms.
Notice:
By rearranging and factor grouping or looking directly for variable ratios with common denominators, often \( x^2 - 6x + c \) structures arise.
Among the multiple responses proposed, notice the correct manipulation might lead away from complex quadratics to:
Types \( a/x + b \) general forms.
By testing:
Testing forms gives way \( 6x \frac{6}{x - 6} \) as a derivative invisible.
The steps simplified further reach:
Picking simplest forms, testing gives us choice relating to zeroes:
Conclusion:
The simplest form representing alignment of variables or required factors can fall to:
\[ \frac{36}{1 - 6x} \]
among \( x - c \) paths leading back simplest when evaluating shifted around common terms.
However, the simplest achieved is indeed:
\[ 36(x - 6x^2 - 1) \]
Yet maintaining speed we see:
Given answer selections respond favor around trait \( = \frac{36}{6} (x - 6)\) leading ends find simple roots or push into other trays.
Conclusively capturing:
The simplest choice can also engage noticing is indeed factoring toward \( x - \frac{6}{x}\) reflecting ratios found among productions of terms divided.
Overall collapsible forms give:
From optioned,
The result forms closer handles \( \frac{20}{x} \) showing factor too, thus settled several forms emergent gave:
We thus answer:
Final answer: \( \frac{20}{x} \)
1 answer