Questions: a) (x+9)^2 x^2+2 cdot x cdot 9+9= x^2+18x+81 b) (3-a)^2= 3-a^2 x C) (x+7) cdot(x-7)= x^2-7 x+7 x^2+7 x= x^2-14 x^2 D) (x+2 y) cdot(x-2 y)= x^2-2 y x+2 y x-4 y^2 e) (3 y^2-2)^2=

Transcript text: a) \((x+9)^{2}\) \[ x^{2}+2 \cdot x \cdot 9+9= \\ x^{2}+18 x+81 \] b) \[ (3-a)^{2}= \\ 3-a^{2} x \] C) \[ (x+7) \cdot(x-7)= \\ x^{2}-7 x+7 x^{2}+7 x= \\ x^{2}-14 x^{2} \] D) \[ (x+2 y) \cdot(x-2 y)= \\ x^{2}-2 y x+2 y x-4 y^{2} \] e) \(\left(3 y^{2}-2\right)^{2}=\)

Solution

Solution Steps

Solution Approach

a) Expand the binomial \((x+9)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\).

b) Expand the binomial \((3-a)^2\) using the same formula \((a-b)^2 = a^2 - 2ab + b^2\).

c) Expand the product \((x+7)(x-7)\) using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\).

Step 1: Expand \((x+9)^2\)

Using the binomial expansion formula \((a+b)^2 = a^2 + 2ab + b^2\): \[ (x+9)^2 = x^2 + 2 \cdot x \cdot 9 + 9^2 = x^2 + 18x + 81 \] For \(x = 1\): \[ (1+9)^2 = 1^2 + 18 \cdot 1 + 81 = 100 \]

Step 2: Expand \((3-a)^2\)

Using the binomial expansion formula \((a-b)^2 = a^2 - 2ab + b^2\): \[ (3-a)^2 = 3^2 - 2 \cdot 3 \cdot a + a^2 = 9 - 6a + a^2 \] For \(a = 1\): \[ (3-1)^2 = 3^2 - 6 \cdot 1 + 1^2 = 9 - 6 + 1 = 4 \]

Step 3: Expand \((x+7)(x-7)\)

Using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\): \[ (x+7)(x-7) = x^2 - 7^2 = x^2 - 49 \] For \(x = 1\): \[ (1+7)(1-7) = 1^2 - 49 = 1 - 49 = -48 \]