Simplify Using Algebraic Identity Simplify \[ (x^2+y^2-z^2)^2-(x^2-y^2+z^2)^2 \] Solution: Using identity: \[ a^2-b^2=(a-b)(a+b) \] \[ = \left[(x^2+y^2-z^2)-(x^2-y^2+z^2)\right] \left[(x^2+y^2-z^2)+(x^2-y^2+z^2)\right] \] \[ = (2y^2-2z^2)(2x^2) \] \[ = 4x^2(y^2-z^2) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (2x+p-c)^2-(2x-p+c)^2 \] Solution: Using identity: \[ a^2-b^2=(a-b)(a+b) \] \[ = \left[(2x+p-c)-(2x-p+c)\right] \left[(2x+p-c)+(2x-p+c)\right] \] \[ = (2p-2c)(4x) \] \[ = 8x(p-c) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 + (a-b+c)^2 + (a+b-c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (a+b+c)^2 + (a-b+c)^2 + (a+b-c)^2 \] \[ = 3a^2+3b^2+3c^2+2ab-2bc+2ca \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 – (a-b+c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b+c)^2 – (a-b+c)^2 \] \[ = a^2+b^2+c^2+2ab+2bc+2ca \] \[ – (a^2+b^2+c^2-2ab-2bc+2ca) \] \[ = 4ab+4bc \] \[ = 4b(a+c) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 + (a-b+c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b+c)^2 + (a-b+c)^2 \] \[ = a^2+b^2+c^2+2ab+2bc+2ca \] \[ + a^2+b^2+c^2-2ab-2bc+2ca \] \[ = 2a^2+2b^2+2c^2+4ca \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ 4x^2 + y^2 + 25z^2 + 4xy – 10yz – 20zx \] Given: \[ x=4,\quad y=3,\quad z=2 \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ 4x^2 + y^2 + 25z^2 + 4xy – 10yz – 20zx \] \[ = (2x)^2 + y^2 + (5z)^2

Find the Value Using Identity Find the Value \[ a+b+c=9 \] \[ ab+bc+ca=23 \] Find: \[ a^2+b^2+c^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (9)^2 = a^2+b^2+c^2+2(23) \] \[ 81 = a^2+b^2+c^2+46 \] \[ a^2+b^2+c^2 = 81-46 \] \[ =35 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ a^2+b^2+c^2=16 \] \[ ab+bc+ca=10 \] Find: \[ a+b+c \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a+b+c)^2 = 16+2(10) \] \[ (a+b+c)^2 = 16+20 \] \[ (a+b+c)^2 = 36 \] \[ a+b+c = \pm 6 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ a+b+c=0 \] \[ a^2+b^2+c^2=16 \] Find: \[ ab+bc+ca \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (0)^2 = 16+2(ab+bc+ca) \] \[ 0 = 16+2(ab+bc+ca) \] \[ 2(ab+bc+ca) = -16 \] \[ ab+bc+ca = -8 \] Next Question / Full Exercise

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (-2x + 3y + 2z)^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (-2x + 3y + 2z)^2 \] \[ = (-2x)^2 + (3y)^2 + (2z)^2 + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x) \] \[ = 4x^2 + 9y^2 + 4z^2 – 12xy