🧮 algebra

1. The problem is to find the sum of the repeating decimals $0.1\dot{}$ and $0.2\dot{8}$.\n\n2. First, let's express each repeating decimal as a fraction.\n\n3. For $0.1\dot{}$, th

1. **بيان المسألة:** لدينا المتتالية \( (u_n) \) المعرفة ب \( u_0 = 3 \) و \( u_{n+1} = 3u_n - 2 \). المطلوب حساب الحدود \( u_1, u_2, u_3 \). 2. **حساب الحدود:**

1. **State the problem:** Simplify the function $$f(x) = \frac{\sqrt{x-6} + \sqrt{2x-2}}{\sqrt{x-1} - \sqrt{x-5}}.$$\n\n2. **Identify the domain:** For the square roots to be defin

1. The problem is to find the domain of a function, which means determining all possible input values ($x$) for which the function is defined. 2. The domain depends on the type of

1. **State the problem:** Simplify the function $$f(x) = \frac{\sqrt{x-6} + \sqrt{2x-2}}{\sqrt{x-1} - \sqrt{x-5}}.$$\n\n2. **Identify domain restrictions:** The expressions under t

1. **State the problem:** We are given the quadratic function $$f(x) = -2x^2 + 5x + 3$$ and asked to complete the table for the domain $$-2 \leq x \leq 5$$, graph the function incl

1. **Problem statement:** We have a parabola $y=f(x)$ opening downwards passing through the origin and peaking near $(0.5,7)$. A point $P(-2,11)$ lies outside the parabola. We want

1. **State the problem:** Find the product of the radicals in the expression $3\sqrt{2}(5\sqrt{12} - \sqrt{18} + 4\sqrt{24})$. 2. **Recall the formula and rules:**

1. Problem. Solve the quadratic equation $2x^2 - 3x - 5 = 0$.

1. **State the problem:** Find the point $(a,b)$ on the line $2x - 3y + 6 = 0$ that is closest to the point $(3,1)$, then find $a+b$. 2. **Formula and concept:** The closest point

1. The problem asks to find the values of $b$, $c$, and $d$ in the cubic function $y = x^3 + bx^2 + cx + d$ given its graph, and to find the coordinates where the curve crosses the

1. **Stating the problem:** Simplify and evaluate the expression $$12 - (-3) \times \left\{ \frac{1}{4} \div \left[-\frac{1}{2}\right]^2 \right\}$$. 2. **Recall the order of operat

1. **Problema:** Determinare dominio, segno, limiti agli estremi del dominio e asintoti della funzione $$f(x) = \frac{1 - x^3}{4x^2 - 1}$$. 2. **Dominio:** Il denominatore non può

1. **State the problem:** We are given two functions: $$f(x) = \frac{3}{x^2}$$

1. **Problem statement:** Given $a=3.0$, $b=4.24$, and $c=3.654$, find the percentage error of the expression $$\frac{a+b}{c+a}$$. 2. **Formula and explanation:** Percentage error

1. **Problem:** Complete the table by finding the missing values for percentage profit or loss and cost prices. 2. **Formula:** Percentage profit or loss is calculated as $$\text{P

1. **State the problem:** Solve the simultaneous equations using the matrix method: $$2n + 3m = 12$$

1. **State the problem:** A truck contains 15 boxes, each either red or blue. Red boxes weigh 3 kg each, blue boxes weigh 2 kg each. The total weight is 36 kg. We need to find the

1. **Problem:** Simplify the expression $$\sqrt[4]{\sqrt{2}} \cdot \sqrt[3]{4} \cdot \sqrt[5]{32}$$. 2. **Rewrite each radical as an exponent:**

1. **Stating the problem:** Simplify the expression $$\frac{2}{3} \sqrt{18} + 2 \sqrt{27} - \sqrt{108} + 0.3 \sqrt{200}$$. 2. **Recall the rule:** Simplify square roots by factorin

1. **Problem statement:** Given two functions $f(x) = x^2$ and $g(x) = 2x + 1$, find the following combined functions: $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\frac{f}{g}(x)$. 2. *