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the expression $x(x-9)$ is resolved in the manner $$x(x-9)=x^2-9x$$ but why not like this? $$x(x-9)=x^2-9$$ where does the $9x$ come from?

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Remember the distributive law $$ a(b+c) = ab + ac . $$

In your expression, $a=x$, $b=x$ and $c = -9$.

Then $x$ times $-9$ is rewritten as $-9x$.

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$$x(x-9)$$ By the Distributive Law $$x^2-9x$$

Here's a short "proof" to show that you can't derive $x^2-9$ from $x(x-9)$.

$$\text{Proof by Contradiction}$$

Assume that

$$x(x-9) = x^2-9$$

Further, we will say that

$$f(x)=x(x-9)$$ and $$g(x) = x^2-9$$

If $$x(x-9) = x^2-9$$

then $$f(x)=g(x)$$

Because $$f(x)=g(x), f(0) = g(0)$$

This, however, is not the case

$$0(0-9) \neq 0^2-9$$

$$ \therefore f(x) \neq g(x)$$

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  • $\begingroup$ Your "short proof" could be just one line long: the equation fails for $x=0$. $\endgroup$ – Ethan Bolker Jul 28 '19 at 2:09
  • $\begingroup$ Ha, note taken. $\endgroup$ – N. Bar Jul 28 '19 at 2:15
  • $\begingroup$ You don't need all that stuff defining $f$ and $g$. Just the next to last line. $\endgroup$ – Ethan Bolker Jul 28 '19 at 2:19
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What's the area of the dark green rectangle? The area of the light green rectangle is $A = x(x-9)$, but it's also the difference between the square $x^2$ and the dark green rectangle. What is this difference? Can you see where the distributive property [e.g. $a(b-c) = ab - ac$] comes from?

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