Question
Download Solution PDFIf (a + b - c) = 20, and a2+ b2 + c2 = 152, find the value of a3 + b3 - c3 + 3abc.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
(a + b - c) = 20;
a2+ b2 + c2 = 152
Formula used:
a3 + b3 = (a + b)3 - 3ab × (a + b)
Calculation:
We have 3 variables in this equation and 2 equations are given.
then let c = 0
⇒ (a + b) = 20
⇒ a2 + b2 = 152
Now,
(a + b) = 20
Squaring both sides
⇒ a2 + b2 + 2ab = 400
⇒ 152 + 2ab = 400
⇒ 2ab = 400 - 152 = 248
⇒ ab = 248/2 = 124
a3 + b3 = (a + b)3 - 3ab × (a + b)
⇒ (20)3 - 3 × 124 × 20
⇒ 8000 - 7440 = 560
∴ The correct answer is 560.
Alternate MethodConcept:
Use algebraic identities to find the required expression.
Calculation
Use the identity for the sum of cubes:
We know,
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
if c ⇒ - c, then
a3 + b3 - c3 + 3abc = (a + b - c){(a2 + b2 + c2 - ab - b(-c) - (-c)a)}
⇒ a3 + b3 - c3 + 3abc = (a + b - c)(a2 + b2 + c2 - ab + bc + ca)
⇒ a3 + b3 - c3 + 3abc = (a + b - c){(a2 + b2 + c2 - (ab - bc - ca)}
Given:
a + b - c = 20
a2 + b2 + c2 = 152
Find ( ab + bc - ca):
⇒ (a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca
⇒ 202 = 152 + 2ab - 2bc - 2ca
⇒ 400 = 152 + 2(ab - bc - ca)
⇒ 248 = 2(ab - bc - ca)
⇒ ab - bc - ca = 124
Substitute in the expression:
a3 + b3 - c3 + 3abc
⇒(a + b - c){(a2 + b2 + c2 - (ab - bc - ca)}
= 20 × (152 - 124)
= 20 × 28
= 560
∴ The value of a3 + b3 - c3 + 3abc is 560.
Last updated on Jun 13, 2025
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