1. **Problem statement:** We will rewrite and reduce expressions using the square formulas (kvadratsætningerne). 2. **Square formulas used:** - $(a+b)^2 = a^2 + 2ab + b^2$ - $(a-b)^2 = a^2 - 2ab + b^2$ - $(a+b)(a-b) = a^2 - b^2$ --- ### Opgave 1: Rewrite using square formulas 1) $(x+y)^2 = x^2 + 2xy + y^2$ 2) $(3x+4y)^2 = (3x)^2 + 2\cdot 3x \cdot 4y + (4y)^2 = 9x^2 + 24xy + 16y^2$ 3) $(8+2a)^2 = 8^2 + 2 \cdot 8 \cdot 2a + (2a)^2 = 64 + 32a + 4a^2$ 4) $(n-m)^2 = n^2 - 2nm + m^2$ 5) $(5x - y)^2 = (5x)^2 - 2 \cdot 5x \cdot y + y^2 = 25x^2 - 10xy + y^2$ 6) $(10 - 4g)^2 = 10^2 - 2 \cdot 10 \cdot 4g + (4g)^2 = 100 - 80g + 16g^2$ 7) $(x+y)(x-y) = x^2 - y^2$ 8) $(4a + b)(4a - b) = (4a)^2 - b^2 = 16a^2 - b^2$ 9) $(10x + 5y)(10x - 5y) = (10x)^2 - (5y)^2 = 100x^2 - 25y^2$ --- ### Opgave 2: Rewrite using square formulas 1) $(5a - 2b)^2 = (5a)^2 - 2 \cdot 5a \cdot 2b + (2b)^2 = 25a^2 - 20ab + 4b^2$ 2) $(2x + y)(2x - y) = (2x)^2 - y^2 = 4x^2 - y^2$ 3) $(4x + 3)^2 = (4x)^2 + 2 \cdot 4x \cdot 3 + 3^2 = 16x^2 + 24x + 9$ 4) $(8x + 3y)(8x - 3y) = (8x)^2 - (3y)^2 = 64x^2 - 9y^2$ 5) $(2n - \frac{1}{2}m)^2 = (2n)^2 - 2 \cdot 2n \cdot \frac{1}{2}m + \left(\frac{1}{2}m\right)^2 = 4n^2 - 2nm + \frac{1}{4}m^2$ --- ### Opgave 3: Reduce expressions using square formulas 1) $(x + y)^2 - (x - y)^2$ Using formulas: $$ (x+y)^2 = x^2 + 2xy + y^2 $$ $$ (x-y)^2 = x^2 - 2xy + y^2 $$ Subtracting: $$ (x+y)^2 - (x-y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy $$ 2) $(2a + b)^2 + (a + b)(a - b) - 4ab$ Calculate each term: $$(2a + b)^2 = 4a^2 + 4ab + b^2$$ $$(a + b)(a - b) = a^2 - b^2$$ Sum: $$4a^2 + 4ab + b^2 + a^2 - b^2 - 4ab = 4a^2 + a^2 + 4ab - 4ab + b^2 - b^2 = 5a^2$$ 3) $(3n - 4m)^2 - (n + 4m)^2 + 32nm$ Calculate squares: $$(3n - 4m)^2 = 9n^2 - 24nm + 16m^2$$ $$(n + 4m)^2 = n^2 + 8nm + 16m^2$$ Subtract and add 32nm: $$9n^2 - 24nm + 16m^2 - (n^2 + 8nm + 16m^2) + 32nm = 9n^2 - 24nm + 16m^2 - n^2 - 8nm - 16m^2 + 32nm$$ Simplify: $$ (9n^2 - n^2) + (-24nm - 8nm + 32nm) + (16m^2 - 16m^2) = 8n^2 + 0 + 0 = 8n^2$$ 4) $(3x + y)(3x - y) - x(9x + 8y) + y^2$ Calculate: $$(3x + y)(3x - y) = 9x^2 - y^2$$ $$- x(9x + 8y) = -9x^2 - 8xy$$ Sum all: $$9x^2 - y^2 - 9x^2 - 8xy + y^2 = -8xy$$ --- **Final answers:** Opgave 1: See above expansions. Opgave 2: See above expansions. Opgave 3: 1) $4xy$ 2) $5a^2$ 3) $8n^2$ 4) $-8xy$