Question

solve using identities ok!!
if x+y=12 and xy=27, find the value of x³+y³​

Answer 1

Answer:

(x+y)³= x³+y³+3xy(x+y)

(12)³=x³+y³+3(27)(12)

1728=x³+y³+972

x³+y³=1728-972

x³+y³=756

Answer 2

Answer:

Given:-

x + y = 12

xy = 27

To find:-

We know that,

${x}^{3} + y {}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )$

$(x + y)[(x + y) {}^{2} - 3xy] \\ \\ =(x + y)[(12) {}^{2} - 3xy]\\ \\ = 12[(12) {}^{2} - 3 \times 27]\\ \\ = 12(144 - 81) \\ \\ = 12 \times 63 \\ \\ = 756$

Thus, x^3 + y^3 = 756.

Some useful indentities:-

→ (a + b)^2 = a^2 + 2ab + b^2

→ (a – b)^2 = a^2 – 2ab + b^2

→ a^2 – b^2 = (a + b) (a – b)

→ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

→ (a + b – c)^2 = a^2 + b^2 + c^2 + 2ab – 2bc – 2ca

→ (a – b – c)^2 = a^2 + b^2 + c^2 – 2ab + 2bc – 2ca

→ (a + b)^3 = a^3 + b^3 + 3ab(a + b)

→ (a – b)^3 = a^3 – b^3 – 3ab(a – b)

→ (a^3 + b^3) = (a + b) (a^2 – ab + b^2)

→ (a^3 – b^3) = (a – b) (a^2 + ab + b^2)