Question

Докажите тождество 2a(a+b-c)-2b(a-b-c)+2c(a-b+c) = 2(a^2+b^2+c^2) Буду очень благодарна за помощь.

Answer 1

2a(a+b-c)-2b(a-b-c)+2c(a-b+c)=

=2a²+2ab-2ac-2ab+2b²+2bc+2ac-2bc+2c²=

=2a²+2b²+2c²=

=2(a²+b²+c²)

Answer 2

Proof of the Identity: 2a(a+b-c)-2b(a-b-c)+2c(a-b+c) = 2(a^2+b^2+c^2)

To prove the given identity: 2a(a+b-c)-2b(a-b-c)+2c(a-b+c) = 2(a^2+b^2+c^2), we will simplify both sides of the equation and show that they are equal.

Left-hand side (LHS): Starting with the left-hand side of the equation: 2a(a+b-c)-2b(a-b-c)+2c(a-b+c)

Expanding the terms: = 2a^2 + 2ab - 2ac - 2ab + 2b^2 + 2bc + 2ac - 2bc + 2c^2

Combining like terms: = 2a^2 + 2b^2 + 2c^2

Right-hand side (RHS): Now let's simplify the right-hand side of the equation: 2(a^2+b^2+c^2)

Since the right-hand side is already simplified, we can directly compare it to the simplified left-hand side.

Comparing LHS and RHS: Comparing the simplified left-hand side (LHS) and right-hand side (RHS), we can see that they are equal: LHS = 2a^2 + 2b^2 + 2c^2 RHS = 2(a^2+b^2+c^2)

Therefore, we have proved that the identity 2a(a+b-c)-2b(a-b-c)+2c(a-b+c) = 2(a^2+b^2+c^2) holds true.

Please let me know if you have any further questions!