Question : If $a + b + c = 12$ and $ab + bc + ca = 22$, then what is the value of $a^3 + b^3 + c^3 - 3abc ?$
Option 1: 1052
Option 2: 936
Option 3: 924
Option 4: 876
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Correct Answer: 936
Solution :
Using the given equation $a + b + c = 12$, we can substitute this value into the formula:
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
⇒ $a^3 + b^3 + c^3 - 3abc = (12)(a^2 + b^2 + c^2 - ab - bc - ca)$
To find the value of $a^2 + b^2 + c^2 - ab - bc - ca$, we'll square the equation $a + b + c = 12$
⇒ $(a + b + c)^2 = 12^2$
⇒ $a^2 + b^2 + c^2 + 2(ab + bc + ca) = 144$
Substituting the given equation $ab + bc + ca = 22$:
$a^2 + b^2 + c^2 + 2(22) = 144$
⇒ $a^2 + b^2 + c^2 = 100$
Now we have the value of $a^2 + b^2 + c^2 - ab - bc - ca$:
$a^2 + b^2 + c^2 - ab - bc - ca = 100 - 22 = 78$
Substituting this value back into the formula, we get:
$a^3 + b^3 + c^3 - 3abc = (12)(78)$
⇒ $a^3 + b^3 + c^3 - 3abc = 936$
Hence, the correct answer is 936.
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