Simplify:
(i) (a+b+c)²+(a-b+c)²
(ii) (a+b+c)²-(a-b+c)²
(iii) (a+b+c)² +(a-b+c)²+(a+b-c)²
(iv) (2x+p+c)²-(2x-p+c)²
(v) (x²+y²-z²)²-(x²-y²+z²)²
Answers
By using an identity (a + b + c)² = a² + b² + c² + 2ab + 2 bc + 2 ca)
(i) (a + b + c)² + (a - b + c)²
= (a² + b² + c² + 2ab + 2bc + 2ca) + (a² + (−b) ² + c²− 2ab − 2bc + 2ca)
= 2a² + 2 b² + 2c² + 4ca
(ii) (a + b + c)² − (a − b + c) ²
= (a² + b² + c² + 2ab + 2bc + 2ca) − (a² + (−b) ² + c²− 2ab − 2bc + 2ca)
= a² + b² + c² + 2ab + 2bc + 2ca − a² − b² − c ²+ 2ab + 2bc − 2ca
= 4ab + 4bc
(iii) (a + b + c)² +(a - b + c)² + (a + b - c)²
= a² + b² + c² + 2ab + 2bc + 2ca + (a² + (− b) ² + c²− 2ab − 2bc + 2ca) + (a² + b² + (- c)² + 2ab − 2bc – 2ab)
= 3a² + 3b² + 3c² + 2ab − 2bc + 2ca
(iv) (2x + p - c)²- (2x - p + c)²
={ (2x)² + p² + (- c)² + 2 × 2x × p + 2 × p (-c) + 2 × (- c) × 2x } - {(2x)² + (-p)² + c² + 2 × 2x × (-p) + 2 × (-p)c + 2 × c × 2x }
= {4x² + p² + c² + 4xp - 2pc - 4xc} − {4x² + p² + c² − 4xp− 2pc + 4xc}
= 4x² + p² + c² + 4xp - 2pc - 4xc − 4x² − p²− c² + 4xp + 2pc − 4cx
= 4xp + 4xp - 4xc - 4xc
= 8xp − 8xc
= 8x(p − c)
(v) (x² + y² - z²)² - (x² - y² + z²)²
= {(x²)² + (y²)² + (−z) ² + 2 × x²y² + 2 y² × - z² + 2 × -z² × x²) } − {(x²)² + (- y²)² + (z) ² + 2 × x² × (-y²) + 2 × (-y²) × z² + 2 × z² × x²) }
= {x⁴ + y⁴ + z⁴ + 2x²y² - 2y²z² - 2x²z² }− {x⁴ + y⁴ + z⁴ − 2x²y² − 2y²z² + 2x²z²}
= x⁴ + y⁴ + z⁴ + 2x²y² - 2y²z² - 2x²z² − x⁴ - y⁴ - z⁴ + 2x²y² + 2y²z² - 2x²z²}
= 2x²y² + 2x²y² - 2x²z² - 2x²z²
= 4x²y² – 4z²x²
HOPE THIS ANSWER WILL HELP YOU…..
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Answer:
Step-by-step explanation:
(i) (a + b + c)² + (a - b + c)²
= (a² + b² + c² + 2ab + 2bc + 2ca) + (a² + (−b) ² + c²− 2ab − 2bc + 2ca)
= 2a² + 2 b² + 2c² + 4ca
(ii) (a + b + c)² − (a − b + c) ²
= (a² + b² + c² + 2ab + 2bc + 2ca) − (a² + (−b) ² + c²− 2ab − 2bc + 2ca)
= a² + b² + c² + 2ab + 2bc + 2ca − a² − b² − c ²+ 2ab + 2bc − 2ca
= 4ab + 4bc
(iii) (a + b + c)² +(a - b + c)² + (a + b - c)²
= a² + b² + c² + 2ab + 2bc + 2ca + (a² + (− b) ² + c²− 2ab − 2bc + 2ca) + (a² + b² + (- c)² + 2ab − 2bc – 2ab)
= 3a² + 3b² + 3c² + 2ab − 2bc + 2ca
(iv) (2x + p - c)²- (2x - p + c)²
={ (2x)² + p² + (- c)² + 2 × 2x × p + 2 × p (-c) + 2 × (- c) × 2x } - {(2x)² + (-p)² + c² + 2 × 2x × (-p) + 2 × (-p)c + 2 × c × 2x }
= {4x² + p² + c² + 4xp - 2pc - 4xc} − {4x² + p² + c² − 4xp− 2pc + 4xc}
= 4x² + p² + c² + 4xp - 2pc - 4xc − 4x² − p²− c² + 4xp + 2pc − 4cx
= 4xp + 4xp - 4xc - 4xc
= 8xp − 8xc
= 8x(p − c)
(v) (x² + y² - z²)² - (x² - y² + z²)²
= {(x²)² + (y²)² + (−z) ² + 2 × x²y² + 2 y² × - z² + 2 × -z² × x²) } − {(x²)² + (- y²)² + (z) ² + 2 × x² × (-y²) + 2 × (-y²) × z² + 2 × z² × x²) }
= {x⁴ + y⁴ + z⁴ + 2x²y² - 2y²z² - 2x²z² }− {x⁴ + y⁴ + z⁴ − 2x²y² − 2y²z² + 2x²z²}
= x⁴ + y⁴ + z⁴ + 2x²y² - 2y²z² - 2x²z² − x⁴ - y⁴ - z⁴ + 2x²y² + 2y²z² - 2x²z²}
= 2x²y² + 2x²y² - 2x²z² - 2x²z²
= 4x²y² – 4z²x²