Factorise the following expressions completely. Leave your answers in simplest form:

These numbers are -1 and -15.

$$(x-1)(x-15)$$

The factorised expression is $\boxed{(x-1)(x-15)}$.

1.1.2 $a(6a + 2b) - 3(3a + b)$

$$(3a+b)(2a-3)$$

The factorised expression is $\boxed{(3a+b)(2a-3)}$.

1.2 Simplify the following expressions completely.

1.2.1 $27^{\frac{2}{3}}$

$$(3)^2 = 9$$

The simplified expression is $\boxed{9}$.

1.2.2 $\frac{x+3}{x-3} \times \frac{x^3-27}{x^2-9} + \frac{x^2+3x+9}{x-3}$

Use difference of cubes $a^3-b^3=(a-b)(a^2+ab+b^2)$ and difference of squares $a^2-b^2=(a-b)(a+b)$.

$$\frac{x+3}{x-3} \times \frac{(x-3)(x^2+3x+9)}{(x-3)(x+3)} + \frac{x^2+3x+9}{x-3}$$

$$\frac{\cancel{x+3}}{\cancel{x-3}} \times \frac{\cancel{(x-3)}(x^2+3x+9)}{\cancel{(x-3)}\cancel{(x+3)}} + \frac{x^2+3x+9}{x-3}$$ $$= \frac{x^2+3x+9}{x-3} + \frac{x^2+3x+9}{x-3}$$

$$= \frac{(x^2+3x+9) + (x^2+3x+9)}{x-3}$$ $$= \frac{2(x^2+3x+9)}{x-3}$$

The simplified expression is $\boxed*{\frac{2(x^2+3x+9)}{x-3}*}$.

1.2.3 $\frac{3^{x+1} - 3^{x+2}}{3^{x+2} + 3^x}$

$$= -\frac{3}{5}$$

The simplified expression is $\boxed*{-\frac{3}{5}*}$.

QUESTION 2

2.1 Solve for x:

2.1.1 $2 \cdot 3^{2x} = 18$

$$3^{2x} = \frac{18}{2}$$ $$3^{2x} = 9$$

$$x = 1$$

The solution is $\boxed{x=1}$.

2.1.2 $\frac{2x-1}{3} \ge 2x+1$

$$-1 - 3 \ge 6x - 2x$$ $$-4 \ge 4x$$

$$\frac{-4}{4} \ge x$$ $$-1 \ge x \quad \text{or} \quad x \le -1$$

The solution is $\boxed{x \le -1}$.

2.1.3 $2x^2 - x - 10 = 0$

Find two numbers that multiply to $2 \times (-10) = -20$ and add to -1. These numbers are -5 and 4.

$$2x^2 - 5x + 4x - 10 = 0$$

$$x(2x-5) + 2(2x-5) = 0$$ $$(2x-5)(x+2) = 0$$

$$2x-5 = 0 \quad \text{or} \quad x+2 = 0$$ $$2x = 5 \quad \text{or} \quad x = -2$$ $$x = \frac{5}{2} \quad \text{or} \quad x = -2$$

The solutions are $\boxed*{x=\frac{5}{2} \text{ or } x=-2*}$.

2.2 How much must be added to $(x+5)(x-5)$ so that it will be equal to $(x+1)(x-25)$?

Set up the equation.

$$(x+5)(x-5) + A = (x+1)(x-25)$$

$$(x^2 - 25) + A = x^2 - 25x + x - 25$$ $$x^2 - 25 + A = x^2 - 24x - 25$$

$$A = (x^2 - 24x - 25) - (x^2 - 25)$$ $$A = x^2 - 24x - 25 - x^2 + 25$$

$$A = -24x$$

The amount that must be added is $\boxed{-24x}$.

2.3 Solve for x and y simultaneously:
Equation 1: $y - 3x = -2$
Equation 2: $7x - 2y = 8$

$$7x - 6x + 4 = 8$$ $$x + 4 = 8$$

$$x = 8 - 4$$ $$x = 4$$

$$y = 3(4) - 2$$ $$y = 12 - 2$$ $$y = 10$$

The solutions are $\boxed*{x=4, y=10*}$.