Question

answer with photos if a + b + C is equal to 9 and a square + b square + c square is equal to 35 find the value of a cube plus b cube plus c cube - 3abc ​

Answer 1

answer is 108.

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Answer 2

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\bf{{a}^{3} + {b}^{3} + {c}^{3} - 3abc = 108}}$

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

Given :- a + b + c = 9, a² + b² + c² = 35

To find :- a³ + b³ + c³ - 3abc

Solution :-

First we need to know the value of ab + bc + ca

$a + b + c = 9$

By squaring on both the sides,

${(a + b + c)}^{2} = {9}^{2}$

We know that, (x + y + z)² = x² + y² + z² + 2(xy + yz + xz)

Here x = a, y = b, z = c

By substituting the values in the identity we have,

${a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ac) = 81$

$( {a}^{2} + {b}^{2} + {c}^{2}) + 2(ab + bc + ac) = 81$

$35 + 2(ab + bc + ca) = 81$

[Since a² + b² + c² = 35]

$2(ab + bc - ac) = 81 - 35$

$ab + bc +ac = \dfrac{46}{2}$

$ab + bc + ac = 23...(1)$

We know that x³ + y³ + z³ - 3xyz = {x + y + z}{x² + y² + z² - (xy + yz + xz)}

Here x = a, y = b, z = c

By substituting the values in the identity we have,

${a}^{3} + {b}^{3} + {c}^{3} - 3abc = (a + b + c)( {a}^{2} + {b}^{2} + {c}^{2} - (ab + bc + ac))$

${a}^{3} + {b}^{3} + {c}^{3} - 3abc = 9(( {a}^{2} + {b}^{2} + {c}^{2}) - 23)$

[Since Given that a + b + c = 9 and ab + bc + ac = 23]

${a}^{3} + {b}^{3} + {c}^{3} - 3abc =9(35 - 23)$

${a}^{3} + {b}^{3} + {c}^{3} - 3abc = 9(12)$

${a}^{3} + {b}^{3} + {c}^{3} - 3abc = 108$

$\boxed{\bf{{a}^{3} + {b}^{3} + {c}^{3} - 3abc = 108}}$

$\mathfrak{\large{\underline{\underline{Identities\:Used:-}}}}$

(x + y + z)² = x² + y² + z² + 2(xy + yz + xz)

x³ + y³ + z³ - 3xyz = {x + y + z}{x² + y² + z² - (xy + yz + xz)}

$\mathfrak{\large{\underline{\underline{Extra\:Information:-}}}}$

1] (x + y)² = x² + y² + 2xy

2] (x - y)² = x² + y² - 2xy

3] (x + y)(x - y) = x² - y²

4] (x + a)(x + b) = x² + (a + b)x + ab

5] x³ + y³ = (x + y)(x² - xy + y²)

6] x³ - y³ = (x - y)(x² + xy + y²)

7] (x + y)³ = x³ + y³ + 3xy(x + y)

8] (x - y)³ = x³ - y³ - 3xy(x - y)