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 : (a + b + c)^2 + (a – b + c)^2 + (a + b – c)^2 Read More »
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 : (a + b + c)^2 – (a – b + c)^2 Read More »
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
Simplify : (a + b + c)^2 + (a – b + c)^2 Read More »
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 of 4x^2 + y^2 + 25z^2 + 4xy – 10yz – 20zx when x = 4, y = 3 and z = 2. Read More »
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
If a + b + c = 9 and ab + bc + ca = 23, find the value of a^2 + b^2 + c^2. Read More »
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
If a^2 + b^2 + c^2 = 16 and ab + bc + ca = 10, find the value of a + b + c. Read More »
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
If a + b + c = 0 and a^2 + b^2 + c^2 = 16, find the value of ab + bc + ca. Read More »
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
Write the following in the expanded form : (-2x + 3y + 2z)^2 Read More »
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
Write the following in the expanded form : (2x – y + z)^2 Read More »
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
Write the following in the expanded form : (x + 2y + 4z)^2 Read More »