Question 93

If a + b + c = 2s, then find the value of $$(s - a)^3 + (s - b)^3 + 3(s - a)(s - b)c.$$

Solution

a + b + c = 2s

$$(s - a)^3 + (s - b)^3 + 3(s - a)(s - b)c$$

$$(s - a)^3 + (s - b)^3 + 3(s - a)(s - b)(s - a + s - b)$$

let x = s-a and y = s - b

$$x^3 + y^3 + 3xy(x + y) = (x + y)^3$$

($$\because (a+b)^3 = a^3 + b^3 + 3ab(a+b)$$)

$$(s - a + s - b)^3 = c^3$$

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