Math, asked by filurazal, 1 year ago

If 3a minus 2b+5c=5 and 6ab +10bc_15ac=14'find the value of 27a cube +125c cube+90abc_8b cube

Answers

Answered by sivaprasath
2
Solution:

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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
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