Question

5) If a2+b2+c2=90 and a+b+c=20. Then
find the value of a3 + b3 + c3-3 abc.​

Answer 1

Answer:

Given:

It is given that a^2 + b^2 + c^2 = 90 and a + b + c = 90.

To Find:

We need to find the value of a^3 + b^3 + c^3 - 3abc.

Solution:

we know

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

Substituting the given values, we have

(20)^2 = 90 + 2(ab + bc + ca)

400 = 90 + 2(ab + bc + ca)

400 - 90 = 2(ab + bc + ca)

310 = 2(ab + bc + ca)

310/2 = (ab + bc + ca)

155 = (ab + bc + ca)

=> (ab + bc + ca) = 155

Now,

a^3 + b^3 + c^3 - 3abc

= [(a + b + c)(a^2 + b ^2 + c^2) - (ab + bc + ca)]

= (20) × (90) - (155)

= 1800 - 155

= 1645

Therefore the value of a^3 + b^3 + c^3 - 3abc is 1645.

Answer 2

Given,

• $\sf{a^{2} + b^{2} + c^{2} = 90}$
• $\sf{a + b + c = 20}$

To Find,

$\boxed{\sf{a^{3} +b^{3} +c^{3} - 3 abc}}$

Solution,

Given that :

$\implies \sf{a + b + c = 20}$

(Squaring both Side)

$\implies \sf{\therefore (a + b + c)^{2} = 20^{2}}$

We know that :

$\boxed{\bold{\red{(a + b + c)^{2} = a^{2} + b^{2} +c^{2} +2(ab + bc + ac)}}}$

$\implies \sf{\therefore a^{2} + b^{2} + c^{2} +2ab + 2bc + 2ac = 400}$

$\implies \sf{90 + 2(ab + bc + ac) = 400}$

(∵Given that a² + b² + c² = 90 )

$\implies \sf{2(ab + bc + ac) = 400 - 90}$

$\implies \sf{ 2(ab + bc + ac)= 310}$

$\implies \sf{ab + bc + ac = \dfrac{310}{2}}$

$\implies \sf{ab + bc + ac = 155}$

We know that :

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

According to the Question Find Value of a³ + b³ + c³ - 3 abc :-

$\implies \sf{20 \times 90 - 155}$

$\implies \sf{1800 \times -155}$

$\implies \boxed{\sf{1645}}$

$\rule{200}2$