Question

☺Solve it ☺

☺ Explanation Needed ☺​

Answer 1

Given :

1. (x - 2y + 3)² + (x + 2y - 3)²
2. (3k - 4r - 2m)² - (3k + 4r - 2m)²
3. (7a - 6b + 5c)² + (7a + 6b - 5c)²

Solution :

• (x - 2y + 3)² + (x + 2y - 3)²

Apply identity : (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

→ [(x)² + (-2y)² + (3)² + 2*x*(-2y) + 2*(-2y)*3 + 2*x*3] + [(x)² + (2y)² + (-3)² + 2*x*2y + 2*2y*(-3) + 2*x*(-3)]

→ [x² + 4y² + 9 - 4xy - 12y + 6x] + [x² + 4y² + 9 + 4xy - 12y - 6x]

→ x² + 4y² + 9 - 4xy - 12y + 6x + x² + 4y² + 9 + 4xy - 12y - 6x

Arranging variables

→ x² + x² + 4y² + 4y² + 9 + 9 - 4xy + 4xy - 12y - 12y + 6x - 6x

→ 2x² + 8y² + 18 - 24y

• (3k - 4r - 2m)² - (3k + 4r - 2m)²

Apply identity : (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

→ [(3k)² + (-4r)² + (-2m)² + 2*3k*(-4r) + 2*(-4r)*(-2m) + 2*3k*(-2m)] - [(3k)² + (4r)² + (-2m)² + 2*3k*4r + 2*4r*(-2m) + 2*3k*(-2m)]

→ [9k² + 16r² + 4m² - 24kr + 16rm - 12km] - [9k² + 16r² + 4m² + 24kr - 16rm - 12km]

→ 9k² + 16r² + 4m² - 24kr + 16rm - 12km - 9k² - 16r² - 4m² - 24kr + 16rm + 12km

Arranging variables

→ 9k² - 9k² + 16r² - 16r² + 4m² - 4m² - 24kr - 24kr + 16rm + 16rm - 12km + 12km

→ - 48kr + 32rm

• (7a - 6b + 5c)² + (7a + 6b - 5c)²

Apply identity : (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

→ [(7a)² + (-6b)² + (5c)² + 2*7a*(-6b) + 2*(-6b)*5c + 2*7a*5c] + [(7a)² + (6b)² + (-5c)² + 2*7a*6b + 2*6b*(-5c) + 2*7a*(-5c)]

→ [49a² + 36b² + 25c² - 84ab - 60bc + 70ac] + [49a² + 36b² + 25c² + 84ab - 60bc - 70ac]

→ 49a² + 36b² + 25c² - 84ab - 60bc + 70ac + 49a² + 36b² + 25c² + 84ab - 60bc - 70ac

Arranging variables

→ 49a² + 49a² + 36b² + 36b² + 25c² + 25c² - 84ab + 84ab - 60bc - 60bc + 70ac - 70ac

→ 98a² + 72b² + 50c² - 12bc

Answer 2

ANSWER✔

✯.SIMPLIFY

✯.(x-2y+3)²+(x+2y-3)²

✯.(3k-4r-2m)² - (3k+4r-2m)²

✯.(7a-6b+5c)² +(7a+6b-5c)²

$\large\underline\bold{IDENTITY \:IN\:USE,}$

$\sf\large\dashrightarrow\bold (a + b + c)^2 = (a)^2 + (b)^2 + (c)^2 + 2ab + 2bc + 2ac$

$\sf\star (x - 2y + 3)^2 + (x + 2y - 3)^2$

$\sf\implies \big( (x)^2 + (-2y)^2 + (3)^2 + 2 \times (x) \times (-2y) + 2 \times (-2y) (3) + 2 \times (x) \times (3) \big)+\big( (x)^2 + (2y)^2 + (-3)^2 + 2 \times (x) \times (2y) + 2 \times (2y) \times (-3) + 2\times(x)\times (-3)$

$\sf\implies \big( x^2 + 4y + 9 - 4xy - 12y + 6x \big) + \big(x^2 + 4y + 9 + 4xy - 12y - 6x \big)$

$\sf\implies x^2 + 4y + 9 - 4xy - 12y + 6x + x^ + 4y + 9 + 4xy - 12y - 6x$

$\sf\implies x^2 + x^2 + 4y + 4y + 9 + 9 - 4xy + 4xy - 12y - 12y + 6x - 6x$

$\sf\implies 2x^2+4y + 4y + 9 + 9 - 4xy + 4xy - 12y - 12y + 6x - 6x$

$\sf\implies 2x^2+8y + 18 - 4xy + 4xy - 12y - 12y + 6x - 6x$

$\sf\implies 2x² + 8y + 18 - 24y$

$\sf{\boxed{\bf{ 2x² + 8y + 18 - 24y}}}$

$\sf\star (3k - 4r - 2m)^2 - (3k + 4r - 2m)^2$

$\sf\implies \big((3k)^2+ (-4r)^2 + (-2m)^2 + 2 \times3k \times (-4r) + 2(-4r)\times(-2m) + 2\times 3k\times(-2m)\big)- \big((3k)^2+ (4r)^2 + (-2m)^2 + 2 \times 3k \times 4r + 2 \times 4r \times (-2m) + 2\times (3k)\times (-2m)]$

$\sf\implies \big(9k^2+ 16r^2 + 4m^2 - 24kr + 16rm - 12km \big) - \big( 9k^2 + 16r^2+ 4m^2 + 24kr - 16rm - 12km \big)$

$\sf\implies 9k^2+ 16r^2 + 4m^2 - 24kr + 16rm - 12km - 9k^2 - 16r^2 - 4m^2 - 24kr + 16rm + 12km$

$\sf\implies 9k^2 - 9k^2 + 16r^2 - 16r^2 + 4m^2 - 4m^2 - 24kr - 24kr + 16rm + 16rm - 12km + 12km$

$\sf\implies \cancel{9k^2} - \cancel{9k^2} + \cancel{16r^2} - \cancel{16r^2} + \cancel{4m^2} - \cancel{4m^2} - 24kr - 24kr + 16rm + 16rm - \cancel{12km} + \cancel{12km}$

$\sf\implies (-48kr+32rm)$

$\sf{\boxed{\bf{\implies (-48kr+32rm) }}}$

$\sf\star (7a - 6b + 5c)^2 + (7a + 6b - 5c)^2$

$\sf\implies \big( (7a)^2 + (-6b)^2 + (5c)^2 + 2 \times7a \times (-6b) + 2 \times (-6b) \times 5c + 2 \times7a \times 5c \big) + \big( (7a)^2 + (6b)^2 + (-5c)^2 + 2 \times7a \times 6b + 2 \times 6b \times (-5c) + 2 \times 7a \times (-5c)\big)$

$\sf\implies \big( 49a^2 + 36b^2 + 25c^2 - 84ab - 60bc + 70ac \big)+ \big( 49a^2 + 36b^2 + 25c^2 + 84ab - 60bc - 70ac]$

$\sf\implies 49a^2 + 36b^2 + 25c^2 - 84ab - 60bc + 70ac + 49a^2 + 36b^2 + 25c^2 + 84ab - 60bc - 70ac$

$\sf\implies 49a^2 + 49a^2 + 36b^2 + 36b^2 + 25c^2 + 25c^2 - \cancel{84ab} + \cancel{84ab} - 60bc - 60bc + \cancel{70ac} - \cancel{70ac}$

$\sf\implies 98a^2 + 72b^2 + 50c^2 - 120bc$

$\sf{\boxed{\bf{ 98a^2 + 72b^2 + 50c^2 - 120bc}}}$

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