# Why is $x(x-9)=x^2-9x$ and not $x^2-9$?

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?

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 $$-9x$$.

$$x(x-9)$$ 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

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

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

• Your "short proof" could be just one line long: the equation fails for $x=0$. – Ethan Bolker Jul 28 '19 at 2:09
• Ha, note taken. – N. Bar Jul 28 '19 at 2:15
• You don't need all that stuff defining $f$ and $g$. Just the next to last line. – Ethan Bolker Jul 28 '19 at 2:19

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?