Issolvi (a-b)^2+(b-c)^2+(a+c)^2-(a-b+c)^2

$$( a - b ) ^ { 2 } + ( b - c ) ^ { 2 } + ( a + c ) ^ { 2 } - ( a - b + c ) ^ { 2 }$$

Evalwa

$$a^{2}+b^{2}+c^{2}$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(p-q\right)^{2}=p^{2}-2pq+q^{2} biex tespandi \left(a-b\right)^{2}.

$$a^{2}-2ab+b^{2}+\left(b-c\right)^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Uża teorema binomjali \left(p-q\right)^{2}=p^{2}-2pq+q^{2} biex tespandi \left(b-c\right)^{2}.

$$a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Ikkombina b^{2} u b^{2} biex tikseb 2b^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Uża teorema binomjali \left(p+q\right)^{2}=p^{2}+2pq+q^{2} biex tespandi \left(a+c\right)^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+a^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Ikkombina a^{2} u a^{2} biex tikseb 2a^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+c^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Ikkombina c^{2} u c^{2} biex tikseb 2c^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a-b+c\right)^{2}$$

Ikkwadra a-b+c.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a^{2}-2ab+2ac+b^{2}-2bc+c^{2}\right)$$

Biex issib l-oppost ta' a^{2}-2ab+2ac+b^{2}-2bc+c^{2}, sib l-oppost ta' kull terminu.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-a^{2}+2ab-2ac-b^{2}+2bc-c^{2}$$

Ikkombina 2a^{2} u -a^{2} biex tikseb a^{2}.

$$a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac+2ab-2ac-b^{2}+2bc-c^{2}$$

Ikkombina -2ab u 2ab biex tikseb 0.

$$a^{2}+2b^{2}-2bc+2c^{2}+2ac-2ac-b^{2}+2bc-c^{2}$$

Ikkombina 2ac u -2ac biex tikseb 0.

$$a^{2}+2b^{2}-2bc+2c^{2}-b^{2}+2bc-c^{2}$$

Ikkombina 2b^{2} u -b^{2} biex tikseb b^{2}.

$$a^{2}+b^{2}-2bc+2c^{2}+2bc-c^{2}$$

Ikkombina -2bc u 2bc biex tikseb 0.

$$a^{2}+b^{2}+2c^{2}-c^{2}$$

Ikkombina 2c^{2} u -c^{2} biex tikseb c^{2}.

$$a^{2}+b^{2}+c^{2}$$

Espandi

$$a^{2}+b^{2}+c^{2}$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(p-q\right)^{2}=p^{2}-2pq+q^{2} biex tespandi \left(a-b\right)^{2}.

$$a^{2}-2ab+b^{2}+\left(b-c\right)^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Uża teorema binomjali \left(p-q\right)^{2}=p^{2}-2pq+q^{2} biex tespandi \left(b-c\right)^{2}.

$$a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Ikkombina b^{2} u b^{2} biex tikseb 2b^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Uża teorema binomjali \left(p+q\right)^{2}=p^{2}+2pq+q^{2} biex tespandi \left(a+c\right)^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+a^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Ikkombina a^{2} u a^{2} biex tikseb 2a^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+c^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Ikkombina c^{2} u c^{2} biex tikseb 2c^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a-b+c\right)^{2}$$

Ikkwadra a-b+c.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a^{2}-2ab+2ac+b^{2}-2bc+c^{2}\right)$$

Biex issib l-oppost ta' a^{2}-2ab+2ac+b^{2}-2bc+c^{2}, sib l-oppost ta' kull terminu.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-a^{2}+2ab-2ac-b^{2}+2bc-c^{2}$$

Ikkombina 2a^{2} u -a^{2} biex tikseb a^{2}.

$$a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac+2ab-2ac-b^{2}+2bc-c^{2}$$

Ikkombina -2ab u 2ab biex tikseb 0.

$$a^{2}+2b^{2}-2bc+2c^{2}+2ac-2ac-b^{2}+2bc-c^{2}$$

Ikkombina 2ac u -2ac biex tikseb 0.

$$a^{2}+2b^{2}-2bc+2c^{2}-b^{2}+2bc-c^{2}$$

Ikkombina 2b^{2} u -b^{2} biex tikseb b^{2}.

$$a^{2}+b^{2}-2bc+2c^{2}+2bc-c^{2}$$

Ikkombina -2bc u 2bc biex tikseb 0.

$$a^{2}+b^{2}+2c^{2}-c^{2}$$

Ikkombina 2c^{2} u -c^{2} biex tikseb c^{2}.

$$a^{2}+b^{2}+c^{2}$$