Question

if 3a - 2b + 5c =5 and 6ab + 10bc - 15ac = 14, find the value of 27a^3 + 125c^3 + 90abc -8b^3

Answer 1

olution:
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Given:

3a-2b+5c = 5,

6ab +10bc-15ac =14
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To find:

27a³ +125c³ +90abc -8b³ = ?
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We can use , 

The identity,.

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

=> (3a-2b+5c)² = (3a)²+ (-2b)²+ (5c)²+ 2(3a)(-2b) + 2(-2b)(5c) + 2(3a)(5c)

As we know,

=> 3a-3b+5c=5,.

=>6ab+10bc-15ac=14

If we substitute it here,

we get,

=>(5)² = 9a² + 4b² + 25c²+2(-6ab) + 2(-10bc)+ 2(15ac)

=>25 = 9a² + 4b² + 25c² -2(6ab +10bc-15ac))

=> 25 = 9a²+ 4b² + 25c² -2(14)

=> 25 = 9a² + 4b² + 25c² -28

=> 25 + 28 = 9a² + 4b² + 25c²

=> 9a² + 4b² +25c² =53 ... (1) 
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We can also tell,

27a³ + 125c³ +90abc -8b³ 

= (3a)³ + (5c)³ + 3(3a)(-2b)(5c) -(2b)³

= (3a)³ + (-2b)³ +(5c)³ + 3(3a)(-3b)(5c) ...(1)
Which is of the form,.
= a³ + b³ + c³ -3abc,.
So, we can use the formula,

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

substituting this formula here,

a = 3a,b = -2b,c = 5c,.

=(3a - 2b+5c)( (3a)² + (-2b)² + (5c)² - (3a)(-2b) - (-2b)(5c) - (3a)(5c) )

= (5)( 9a² + 4b² + 25c² + 6ab +10bc -15ac)

=(5)(53 + 14)

=(5)(67)

=335

  ∴ 27a³ + 125c³ + 90abc -8b³ = 335_____________________________________________________________

                                           Hope it Helps !!