🧮 algebra

1. نبدأ ببيان المشكلة: لدينا متتالية هندسية متناقصة حيث \( v_1, v_2, v_3, v_4, v_5 \) هي حدود المتتالية. 2. نعلم أن المتتالية الهندسية تحقق العلاقة \( v_{n} = v_1 \times r^{n-1} \)

1. **State the problem:** A worker completes tasks 25% faster but must redo 10% of the work due to errors. We want to find the overall efficiency compared to before. 2. **Define va

1. Problem 49: Find the number of intersection points between $y = \sqrt{|x|}$ and $5y = x + 6$. 2. Problem 50: Find the number of intersection points between $y = |x^2 - 4|$ and $

1. **State the problem:** We have two tasks, A and B. Task A alone takes 4 days, and Task B alone takes 6 days. When done together, productivity is reduced by 20% due to coordinati

1. **State the problem:** A water extraction machine starts with a maximum capacity of 80 gallons per hour but loses 10% efficiency every 2 hours. We need to find how many hours it

1. The problem asks to analyze and solve parts related to the conic sections given by the equations: (1) $9x^2 - 16y^2 - 36x - 32y - 124 = 0$ (a homogeneous line equation)

1. **Problem statement:** A principal donates books to teachers. Giving each teacher 5 books leaves 23 books behind. Giving each teacher 6 books requires 8 more books. Find the num

1. نبدأ بكتابة معطيات المسألة: المتتالية \((v_n)\) هندسية وحدودها موجبة تماماً. 2. المعادلات المعطاة:

1. The problem is to evaluate the factorial expression $11!$. 2. Recall that the factorial of a positive integer $n$, denoted $n!$, is the product of all positive integers from 1 t

1. **Problem Statement:** Solve the system of linear equations using Cramer's Rule: $$5x + 2y - 2 = 0$$

1. The problem asks to explain how the term $4x$ appears in the second step when expanding $(x - 2)^2$. 2. The expression $(x - 2)^2$ means $(x - 2)(x - 2)$.

1. **State the problem:** Solve the equation $$\log_{x} 1000 - \log x^{2} + \left(10^{-3}\right)^{0} = 0$$ for $x$. 2. **Recall logarithm properties:**

1. **State the problem:** Solve for $x$ in the equation $$\frac{\sqrt{9x + 27}}{2x + 10} + \frac{\sqrt{9x + 27}}{x + 5} = 1.$$\n\n2. **Simplify the equation:** Notice that $2x + 10

1. نبدأ بتحديد المتتالية الهندسية: إذا كانت الحدود الثلاثة الأولى هي $v_1$, $v_2$, و $v_3$، فإن العلاقة بين الحدود هي $v_2 = v_1 r$ و $v_3 = v_1 r^2$ حيث $r$ هو النسبة المشتركة. 2.

1. **State the problem:** Solve the equation $5(5-7) - 2(3+4) = 0$. 2. **Apply the distributive property:** Multiply inside the parentheses by the numbers outside.

1. **State the problem:** Find the exact values of $x$ such that $$21\log_5(x + 5) - \log_5(2x + 2) = 2.$$\n\n2. **Recall the logarithm properties:**\n- $a\log_b(c) = \log_b(c^a)$\

1. The problem involves simplifying and understanding the expression related to the variables $a$, $b$, $c$, $r$, $B$, $D$, and $\Delta$ given in the form of fractions and products

1. **State the problem:** Simplify and solve the equation $$\frac{a+b}{a \times b} - \frac{a-b}{a \times b} = \frac{b}{8}$$ for $a$ and $b$. 2. **Identify the formula and rules:**

1. **State the problem:** Solve the quadratic equation $x^2 + 25x = 0$. 2. **Formula and rules:** To solve quadratic equations, one common method is factoring. If the equation can

1. **بيان المسألة:** لدينا متتالية حسابية أساسها $r = -5$، ومعلومة أن $u_3^2 + u_7^2 = 650$. المطلوب:

1. Let's state the problem: We want to understand how to work with expressions where the powers of $x$ are different, for example, one term has $x^3$ (cube) and another has $x^2$ (