Quadratic topic- very basic

Question

Express $x(4-x)$ as the difference of two squares.

I do not really quite sure what is meant by difference of two squares.

Answer 1

Use the rule $$(a-b)(a+b)=a^2-b^2.$$ Notice that the r.h.s. is a difference of two squares.

If we can write something as a product of two things, say $pq$, then we can try and find $a$ and $b$ such that $p=a+b$ and $q=a-b$.

Adding these together gives $$ p+q=(a+b)+(a-b)=2a \implies a=\frac{p+q}2. $$ Subtracting them from each other gives $$ p-q=(a+b)-(a-b)=2b\implies b=\frac{p-q}2. $$ So for example with $p=x+1$ and $q=5-x$ it pans out like $$ a=\frac{p+q}2=\frac{(x+1)+(5-x)}2=\frac62=3 $$ and $$ b=\frac{p-q}2=\frac{(x+1)-(5-x)}2=\frac{x+1-5+x}2=\frac{2x-4}2=x-2. $$ So $$ (x+1)(5-x)=pq=a^2-b^2=3^2-(x-2)^2 $$ as a difference of two squares.

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Leaving it to you to do the case $x(4-x)$.

Answer 2

Regrading to @Eric's comment above you can do it also as follows:

$$x(4-x)=4x-x^2=\color{blue}{+4}\color{red}{-4}+4x-x^2=\color{blue}{+4}\color{red}{-}(\color{red}{4}\color{blue}{-}4x\color{blue}{+}x^2)=\color{blue}{2^2}\color{red}{-}(x-2)^2$$