🧮 algebra

1. **State the problem:** Factorise the expression $8x - 6$. 2. **Identify the common factor:** Both terms $8x$ and $6$ have a common factor of $2$.

1. **State the problem:** We are given two functions:

1. **State the problem:** Factorise the expression $8x - 6$. 2. **Identify the common factor:** Both terms $8x$ and $6$ have a common factor of $2$.

1. **State the problem:** Factorise the expression $8x - 6$. 2. **Recall the factoring rule:** To factorise an expression, find the greatest common factor (GCF) of all terms and fa

1. **State the problem:** We are given the profit function $$P(x) = 850x - 0.1x^2 - 4000$$ where $x$ is the number of units produced and sold.

1. **State the problem:** Solve the equation $$2000 \times (1.02)^x = 2500$$ for $x$. 2. **Isolate the exponential term:** Divide both sides by 2000 to get:

1. **State the problem:** Fully factorise the expression $8p + 12$. 2. **Identify the greatest common factor (GCF):** The coefficients are 8 and 12. The GCF of 8 and 12 is 4.

1. **Problem statement:** Given nonnegative real numbers $a,b,c$ satisfying $$(a + b)(b + c)(c + a) = 2,$$ prove that $$(a^2 + bc)(b^2 + ca)(c^2 + ab) + a^2 b^2 c^2 \leq 1.$$ 2. **

1. **State the problem:** We need to find the difference quotient $$\frac{f(x+h)-f(x)}{h}$$ for the function $$f(x) = -x^2 + 6x + 3$$ where $$h \neq 0$$. 2. **Write the formula:**

1. **State the problem:** Given the function $f(x) = 2x^2 + 4$ and the equation $f(x + 2) = 2x^2$, find the value of $x$. 2. **Use the function definition:** We know that $f(x + 2)

1. **State the problem:** We are given the function $f(x) = \frac{13}{x}$ and need to simplify the difference quotient:

1. **State the problem:** We need to simplify the difference quotient for the function $f(x) = \sqrt{15x}$, which is given by $$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$$

1. **State the problem:** Simplify the expression $-2^{2}+4[16-:(3-5)]$. 2. **Understand the order of operations:** We follow PEMDAS (Parentheses, Exponents, Multiplication and Div

1. **State the problem:** Solve the system of equations: $$\begin{cases} y = 2x + 1 \\ x + z = 3 \\ x + y + z = 8 \end{cases}$$

1. **State the problem:** Solve the equation $1 (2 - y)^2 = -3y (2 - y)$ for $y$. 2. **Rewrite the equation:**

1. **State the problem:** Simplify the expression $$\left( \frac{4x^{-3} y^{4}}{8x^{2} y^{-2}} \right)^{-2}$$. 2. **Apply the quotient rule for exponents:** When dividing like base

1. Planteamos el problema: Encontrar la función lineal para el ejercicio 17. 2. Ejercicio 17: Dada la pendiente $m=\frac{1}{4}$ y la ordenada al origen $b=-3$, la forma punto-pendi

1. The problem is to understand how Step 3 can be applied to all similar equations. 2. Step 3 usually involves simplifying or manipulating the equation using algebraic rules such a

1. **State the problem:** Simplify the expression $$2\sqrt{t} + \frac{2t + 1}{2\sqrt{t}}$$ and verify the given simplifications. 2. **Recall the rules:**

1. **Problem Statement:** Find the gradient (slope) and y-intercept of the lines given by the graphs (a) and (b). 2. **Formula for slope (gradient):**

1. **Problem statement:** Estimate the amount Ann will receive when exchanging money using the given graphs.