May 2026

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

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}