Kvadratsetningene

$$\begin{array}{l}=(x^2+2·x·2+2^2)+(x^2-2·x·3+3^2)+(x^2-2^2)\\ \\ =(x^2+4x+4)+(x^2-6x+9)+(x^2-4)\\ \\ =x^2+4x+4+x^2-6x+9+x^2-4\\ \\ =3x^2-2x+9\end{array}$$

$$\begin{array}{l}=(16-a^2)-(a^2+4a+4)+(a^2-4a+4)\\ \\ =16-a^2-a^2-4a-4+a^2-4a+4\\ \\ =-a^2-8a+16\end{array}$$

$$\begin{array}{l}=-3(2^2-2·2·x)-5\left(\right(3x{)}^2-2·3x·4+4^2)+2x^2\\ \\ =-3(4-4x+x^2)-5(9x^2-24x+16)+2x^2\\ \\ =-(12-12x-3x^2)-(45x^2-120x+80)+2x^2\\ \\ =12+12x-3x^2-45x^2+120x-80+2x^2\\ \\ =-46x^2+132x-92\end{array}$$

$$\begin{array}{l}=-(4^2-x^2)-\left(\right(2x{)}^2-2·\left(2x\right)·5+5^2)-(x^2-2·x·\frac{1}{3}+\left(\frac{1}{3}{)}^2\right)\\ \\ =-(16-x^2)-(4x^2-20x+25)-(x^2-\frac{2x}{3}+\frac{1}{9})\\ \\ =-16+x^2-4x^2+20x-25-x^2+\frac{2x}{3}-\frac{1}{9}\\ \\ =-41-4x^2+20x+\frac{2x}{3}-\frac{1}{9}\\ \\ =\frac{41·9}{1·9}-4x^2+\frac{20x·3}{1·3}+\frac{2x}{3}-\frac{1}{9}\\ \\ =\frac{369}{9}-4x^2+\frac{60x}{3}+\frac{2x}{3}-\frac{1}{9}\\ \\ =-4x^2+\frac{62}{3}x-\frac{370}{9}\end{array}$$

$$\begin{array}{l}=4-a^2-2(a^2+2a+1)-9(a^2-\frac{2}{3}a+\frac{1}{9})\\ \\ =4-a^2-2a^2-4a-2-9a^2+\frac{9·2}{3}a-\frac{9·1}{9}\\ \\ =4-a^2-2a^2-4a-2-9a^2+6a-1\\ \\ =-12a^2+2a+1\end{array}$$

$$\begin{array}{l}=2(4a^2-1)-(a^2-4a+4)-4(a^2+a+\frac{1}{4})\\ \\ =8a^2-2-a^2+4a-4-4a^2-4a-1\\ \\ =3a^2-7\end{array}$$

$$\begin{array}{l}=a^2-a+\frac{1}{4}+\frac{1}{4}(4a^2+4a+1)-\frac{1}{2}(4a^2-1)\\ \\ =a^2-a+\frac{1}{4}+a^2+a+\frac{1}{4}-2a^2+\frac{1}{2}\\ \\ =1\end{array}$$

$$\begin{array}{l}=(\sqrt{2}{)}^2-1\\ \\ =2-1\\ \\ =1\end{array}$$

$$\begin{array}{l}=(\sqrt{5}{)}^2-2^2\\ \\ =5-4\\ \\ =1\end{array}$$

$$=(30-1)(30+1)={30}^2-1^2=900-1=899$$

$$=(20-2)(20+2)={20}^2-2^2=400-4=396$$

$$=(20+5)(20-5)={20}^2-5^2=400-25=375$$

$$=(100+3)(100-3)={100}^2-3^2=10000-9=9991$$