🧮 algebra

1. **Problem statement:** The first, fifth, and eighth terms of an arithmetic progression (AP) with common difference $3$ form the first three terms of a geometric progression (GP)

1. **State the problem:** We have an arithmetic progression (AP) where the first term $a_1 = 2$.

1. **State the problem:** Solve the complex equation $$p(z) = z^2 - 3z^2 + z + 5$$ for $z$. 2. **Simplify the polynomial:** Combine like terms:

1. **State the problem:** Find the intersection point of the system of equations: $$\begin{cases} 2x + y = 8 \\ 2x + 4y = 20 \end{cases}$$

1. The problem is to analyze and understand the function $$y=2-\sqrt{2-x}$$ and describe its graph. 2. The function involves a square root, so the expression inside the root must b

1. **Problem statement:** Simplify the expression $\sqrt{55 + 30\sqrt{2}}$. 2. **Formula and approach:** We try to express the expression inside the square root in the form $(a + b

1. **State the problem:** We need to find the value of $f(-2 - g(3))$ given the functions $f(x) = \frac{3}{4}x + 10$ and $g(x) = x^2 - 3$. 2. **Calculate $g(3)$:** Substitute $x=3$

1️⃣ Factoring helps us find where a graph crosses the $x$-axis. 🌈 2️⃣ Imagine the equation $y = x^2 - 5x + 6$. We want to find where $y=0$.

1. **بیان مسئله:** عبارت درجه دوم $x^2 + 5x + 9$ را تجزیه کنید. 2. **فرمول تجزیه:** برای یک چندجمله‌ای درجه دوم به شکل $ax^2 + bx + c$، به دنبال دو عدد هستیم که حاصل‌ضربشان $ac$ و

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 9$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** Simplify the expression $$-5^2 + (-5)^2$$. 2. **Recall the order of operations and exponent rules:**

1. **Stating the problem:** Fathi has 1.10 in his printing account. Each sheet of paper costs 0.25. He wants to print a 47-page PDF. He prints on both sides and 2 pages per side.

1. 問題陳述： 假設一個長方形的兩邊分別落在 x 軸和 y 軸上，另一頂點落在直線 $$2x + y = 12$$ 上，求此長方形的最大面積。

1. **Stating the problem:** We are given the system of linear equations:

1. **Problem:** Given that $$\sum_{r=1}^k r^3 = \frac{k^2 (k+1)^2}{4}$$ and $$k$$ is a positive integer, find $$\sum_{r=3}^k (r-1)(r^2 + r + 1)$$ in terms of $$k$$. 2. **Step 1: Si

1. **Problem statement:** Solve the equation $$(x - 2)(x - 3) = 20$$ for $x$. 2. **Formula and rules:** To solve this, first expand the left side using the distributive property (F

1. **State the problem:** Solve the equation $-7(8b-2)=32+3b$ for $b$. 2. **Apply the distributive property:** Multiply $-7$ by each term inside the parentheses:

1. **State the problem:** We are given the system of equations: $$ (x-1)^2 + y^2 = 2 $$

1. The problem is to understand the concepts of GCD (Greatest Common Divisor) and LCM (Least Common Multiple) of two numbers. 2. The GCD of two integers is the largest positive int

1. The problem is to find the value of an expression or function when $X=200$ and $Y=10$. 2. Since the original expression or function is not provided, let's assume a general form

1. **Problem Statement:** We want to determine which XP formula is more effective for reaching level 40. The formulas calculate total XP needed based on base XP per level $X$ and a