Question

if a^2+b^2+c^2=250 and a+b+c=25 find ab+bc+ac​

Answer 1

Given :-

• a² + b² + c² = 250
• a + b + c = 25

Aim :-

• To find the value of ab + bc + ca

Identity to use :-

(a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca => a² + b² + c² + 2(ab + bc + ca)

Substituting a² + b² + c² for 250 and a + b + c as 25,

(25)² = 250 + 2(ab + bc + ca)

625 = 250 + 2(ab + bc + ca)

375 = 2(ab + bc + ca)

375 ÷ 2 = ab + bc + ca

187.5 = ab + bc + ca

Verification :-

Substituting ab + bc + ca for 187.5,

(a+b+c)² = a² + b² + c² + 2(ab + bc + ca)

(25)² = 250 + 2(187.5)

625 = 250 +  375

LHS = RHS

Therefore we can conclude that ab + bc + ca = 187.5

Some more identities :-

• (a+b)² = a² + 2ab + b²

• (a-b)² = a² - 2ab + b²

• (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca

• (x+a)(x+b) = x² + x(a+b) + ab

• a²-b² = (a+b)(a-b)

• (a+b)³ = a³ + 3a²b + 3ab² + b³

• (a-b)³ = a³ - 3a²b + 3ab² - b³

• a³+b³ = (a+b)(a² - ab + b²)

• a³-b³ = (a-b)(a² + ab + b²)

• a³+b³+c³ - 3abc = (a+b+c)(a² + b² + c² - ab - bc - ca)

Conditional identity :-

if a + b + c = 0,

then,

• a³ + b³ + c³ = 3abc