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

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

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (x + 2y + 4z)^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (x + 2y + 4z)^2 \] \[ = x^2 + (2y)^2 + (4z)^2 + 2(x)(2y) + 2(2y)(4z) + 2(4z)(x) \] \[ = x^2 + 4y^2 + 16z^2 + 4xy

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \] \[ = \left(\frac{a}{bc}\right)^2 + \left(\frac{b}{ca}\right)^2 + \left(\frac{c}{ab}\right)^2 + 2\left(\frac{a}{bc}\right)\left(\frac{b}{ca}\right) + 2\left(\frac{b}{ca}\right)\left(\frac{c}{ab}\right) + 2\left(\frac{c}{ab}\right)\left(\frac{a}{bc}\right) \] \[ = \frac{a^2}{b^2c^2} + \frac{b^2}{c^2a^2} + \frac{c^2}{a^2b^2} + \frac{2}{c^2}

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \] \[ = \left(\frac{x}{y}\right)^2 + \left(\frac{y}{z}\right)^2 + \left(\frac{z}{x}\right)^2 + 2\left(\frac{x}{y}\right)\left(\frac{y}{z}\right) + 2\left(\frac{y}{z}\right)\left(\frac{z}{x}\right) + 2\left(\frac{z}{x}\right)\left(\frac{x}{y}\right) \] \[ = \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + \frac{2x}{z}

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

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (a^2 + b^2 + c^2)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ (a^2 + b^2 + c^2)^2 \] \[ = (a^2)^2 + (b^2)^2 + (c^2)^2 + 2(a^2)(b^2) + 2(b^2)(c^2) + 2(c^2)(a^2) \] \[ = a^4 + b^4 + c^4 + 2a^2b^2

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