🧮 algebra

1. مسئله: داده شده دو تابع به صورت مجموعه‌ای از زوج‌های مرتب: $$g = \{(-1,2),(0,3),(2,4),(3,0)\}$$

1. مسئله: دامنه توابع $\frac{f}{g}$ و $f-g$ را پیدا کنید و تابع $\frac{f}{g}$ را به صورت زوج مرتب بنویسید. 2. تعریف دامنه: دامنه تابع $f-g$ شامل تمام نقاطی است که در هر دو تابع $f$

1. **State the problem:** We have a set of points $f = \{(a,2), (2,5), (-2,4), (a^2 + a, 5)\}$ and need to find values of $a$ such that $f$ is a function. A function assigns exactl

1. **Problem (الف): Solve the equation** $$\frac{2x}{x^2 - 1} + \frac{2}{x + 1} = \frac{2 - x}{x^2 - x}$$

1. **State the problem:** Simplify the expression $15 + 9 \div 3$. 2. **Recall the order of operations:** Division and multiplication are performed before addition and subtraction.

1. Задача: минимизировать функцию многих переменных $$f(x,y)=(x-3)^2+(y-2)^2+5$$. 2. Формула: функция квадратичная, минимум достигается в точке, где частные производные равны нулю.

1. **State the problem:** Calculate the sum of $\frac{1}{2}$ and $\frac{1}{3}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. The sum of two frac

1. **بيان المسألة:** لدينا المتجهات:

1. The problem is to analyze the function $h(x) = 3|x - 2|$. 2. The absolute value function $|x - 2|$ measures the distance of $x$ from 2 on the number line.

1. **Problem Statement:** We are given that the sum of the first $n$ terms of two arithmetic progressions (APs) are in the ratio $3n + 31 : 5n - 3$. We need to find the ratio of th

1. **Problem Statement:** Given an arithmetic progression (AP), the $p$th term is $q$, the $q$th term is $p$. We need to find the $(p+q)$th term. 2. **Formula for the $n$th term of

1. **State the problem:** Simplify the expression $2(q + 6)$ using the distributive property. 2. **Formula used:** The distributive property states that $a(b + c) = ab + ac$.

1. **State the problem:** Simplify the expressions: $$A = \ln\left(\frac{1}{3}\right) - \ln(27) + \ln\left(8 \frac{1}{2}\right)$$

1. **State the problem:** Simplify the expression $$\frac{8a}{9c} - \frac{64a^2 + 81c^2}{72ac} + \frac{9c - 64a}{8a}.$$\n\n2. **Identify the common denominator:** The denominators

1. **State the problem:** Simplify the expression $$\left(\frac{1}{5a} + \frac{1}{7a}\right) \times \frac{a^2}{4}$$. 2. **Find a common denominator for the sum inside the parenthes

1. **State the problem:** Simplify the expression $X^2 - 4$. 2. **Formula and rules:** Recognize that $X^2 - 4$ is a difference of squares, which follows the formula:

1. **State the problem:** Simplify the expression $$\frac{10x^8}{2x^5}$$. 2. **Recall the division rule for exponents:** When dividing powers with the same base, subtract the expon

1. نبدأ بكتابة المعادلة المعطاة: $$8 = 2x - x^2$$. 2. نعيد ترتيب المعادلة لجعلها مساوية للصفر: $$-x^2 + 2x - 8 = 0$$.

1. **State the problem:** Simplify or analyze the expression $x^2 - \sqrt{x}$. 2. **Recall the definitions:** $x^2$ means $x$ multiplied by itself, and $\sqrt{x}$ means the square

1. Planteamos el problema para la expresión (e): $$\frac{3^{-n} + \frac{1}{2n}}{\frac{1}{6^7} \left(\frac{7}{6}\right)^{n-1}} \cdot \frac{5^n}{2^n + 3^n}$$

1. The problem is to factor a quadratic expression using the product-sum method. 2. The product-sum method involves finding two numbers that multiply to the product of the quadrati