| (1) | $$ \begin{align} (x-3)^2+(x-1)(x+7) &= (x^2-6x+9)+(x^2+6x-7) \\ &= 2x^2+2 \end{align}$$ |
| (2) | $$ \begin{align} (x+2)(x+9)-x(x+10) &= (x^2+11x+18)-(x^2+10x) \\ &= x^2+11x+18-x^2-10x \\ &= x+18 \end{align}$$ |
| (1) | $$ \begin{align} (x+7)(x+4) &= x^2+11x+28 \end{align}$$ |
| (2) | $$ \begin{align} (x+10)(x-2) &= x^2+8x-20 \end{align}$$ |
| (3) | $$ \begin{align} (x-2)(x+8) &= x^2+6x-16 \end{align}$$ |
| (4) | $$ \begin{align} (x-4y)(x-9y) &= x^2-13xy+36y^2 \end{align}$$ |
| (5) | $$ \begin{align} (x+4)^2 &= x^2+8x+16 \end{align}$$ |
| (6) | $$ \begin{align} (3x-2)^2 &= 9x^2-12x+4 \end{align}$$ |
| (7) | $$ \begin{align} (4x-3y)^2 &= 16x^2-24xy+9y^2 \end{align}$$ |
| (8) | $$ \begin{align} \left( \frac{1}{2}x+2 \right)^2 &= \left(\frac{1}{2}x\right)^2+x\times 2+2^2 \\ &= \frac{1}{4}x^2+2x+4 \end{align}$$ |
| (9) | $$ \begin{align} (x+1)(x-1) &= x^2-1 \end{align}$$ |
| (10) | $$ \begin{align} (x-7x)(x+7y) &= x^2-49y^2 \end{align}$$ |
| (1) | $$ \begin{align} \left(x+\frac{2}{3}\right) \left(x-\frac{1}{3}\right) &= x^2+\left(\frac{2}{3}-\frac{1}{3}\right)x+\frac{2}{3}\times \left(-\frac{1}{3}\right) \\ &= x^2+\frac{1}{3}x-\frac{2}{9} \end{align}$$ |
| (2) | $$ \begin{align} \left(a-\frac{1}{2}\right) \left(a-\frac{1}{4}\right) &= a^2+\left(-\frac{1}{2}-\frac{1}{4}\right)x+\left(-\frac{1}{2}\right)\times \left(-\frac{1}{4}\right) \\ &= a^2-\frac{3}{4}x+\frac{1}{8} \end{align}$$ |
| (3) | $$ \begin{align} (1-x)^2 &= 1-2x+x^2 \end{align}$$ |
| (4) | $$ \begin{align} (5-t)(5+t) &= 25-t^2 \end{align}$$ |
| (5) | $$ \begin{align} (-5x+1)(5x+1) &= (1-5x)(1+5x) \\ &= 1-25x^2 \end{align}$$ $$ \left[ \begin{align} (-5x+1)(5x+1) &= -25x^2+1 \end{align} \right]$$ |
| (6) | $$ \begin{align} \left(2x+\frac{1}{2}y\right) \left(2x-\frac{1}{2}y\right) = 4x^2-\frac{1}{4}y^2 \end{align}$$ |
| (1) | $$ \begin{align} (x-7)(x+7)-(x-6)^2 &= (x^2-49)-(x^2-12x+36) \\ &= x^2-49-x^2+12x-36 \\ &= 12x-85 \end{align}$$ |
| (2) | $$ \begin{align} (x+1)(x+5)+(x-2)(x-4) &= (x^2+6x+5)+(x^2-6x+8) \\ &= x^2+6x+5+x^2-6x+8 \\ &= 2x^2+13 \end{align}$$ |
| (3) | $$ \begin{align} (x+2)(x+3)-(x-6)(x+1) &= (x^2+5x+6)-(x^2-5x-6) \\ &= x^2+5x+6-x^2+5x+6 \\ &= 10x+12 \end{align}$$ |
| (4) | $$ \begin{align} (a+b)^2-(a-b)^2 &= (a^2+2ab+b^2)-(a^2-2ab+b^2) \\ &= a^2+2ab+b^2-a^2+2ab-b^2 \\ &= 4ab \end{align}$$ |
| (5) | $$ \begin{align} (2x+y)^2-(x-3y)(x+3y) &= (4x^2+4xy+y^2)-(x^2-9y^2) \\ &= 4x^2+4xy+y^2-x^2+9y^2 \\ &= 3x^2+4xy+10y^2 \end{align}$$ |