🧮 algebra

1. **Problem:** Determine the domain, intercepts, asymptotes, and sketch the graph of $$f(x) = \frac{x^2 + 3x + 2}{x^2 - 1}$$. 2. **Formula and rules:**

1. **State the problem:** Given the cubic equation $$x^3 - 9x^2 + 23x - 15 = 0$$ with roots $$\alpha, \beta, \gamma$$, find the value of $$\frac{1}{\alpha+2} + \frac{1}{\beta+2} +

1. **State the problem:** Simplify the expression $20x - 6y + 2 - 12x + 8y - 10 + 15$. 2. **Group like terms:** Combine the terms with $x$, the terms with $y$, and the constant ter

1. **הבעיה:** מצא את משוואת הישר AB, הישר המקביל לו שעובר דרך נקודה C, והישר המאונך לו שעובר דרך נקודה C. 2. **נתונים:**

1. **Problem statement:** We want to find the number of functions $f:\{1,2,\ldots,10\} \to \{1,2,\ldots,10\}$ such that $f(f(i))=i$ for all $i \in \{1,2,\ldots,10\}$. 2. **Understa

1. The problem is to prove a given relation without finding the value of any variable. 2. To prove a relation, we start by stating the relation clearly and then use algebraic ident

1. The problem is to find the value of the expression or equation that your textbook shows as -6. 2. Since you did not provide the exact expression, I will explain a common scenari

1. **State the problem:** Solve the equation $8 - \frac{1}{4} = \frac{24}{4} - \frac{1}{4}$. 2. **Rewrite the whole numbers as fractions:**

1. **State the problem:** Simplify the expression $$-\cos^4 x + 4 \sin x - \frac{4}{3} \sin^3 x$$ and check if it equals $$\frac{1}{4} \cos^4 x$$. 2. **Recall identities:** Use the

1. **Stating the problem:** A notebook costs floui times the price of a pencil. The pencil costs 30p less than the notebook. We need to find the price of the notebook. 2. **Define

1. **State the problem:** Solve the equation $$-\frac{25}{16}a = -\frac{7}{4}$$ for $a$. 2. **Formula and rules:** To solve for $a$, we need to isolate $a$ by dividing both sides o

1. **State the problem:** Joe and Frank have the same amount of money initially. We want to find how much Joe must give Frank so that Frank has 10 cedis more than Joe. 2. **Define

1. **State the problem:** Solve the equation $$2 \arcsin(x^2 - 3x + 1) + \pi = 0$$ and find the sum of all values of $x$ that satisfy it. 2. **Isolate the arcsin term:**

1. Bài toán yêu cầu tính tổng tất cả các phần tử của ma trận nghịch đảo $A^{-1}$ với ma trận $A = \begin{bmatrix} 2 & 9 \\ 3 & 13 \end{bmatrix}$.\n\n2. Công thức tính ma trận nghịc

1. **State the problem:** Solve the equation $$-\frac{25}{16}a = -\frac{7}{4}$$ for $a$. 2. **Formula and rules:** To solve for $a$, we need to isolate $a$ by dividing both sides o

1. Bài toán yêu cầu tìm giá trị của $m$ để ma trận $$A = \begin{bmatrix} 1 & 0 & 3 \\ m & 5 & 3 \\ 2 & 0 & m \end{bmatrix}$$ không khả nghịch. 2. Một ma trận vuông không khả nghịch

1. The problem is to solve the equation given by the user. However, no specific equation was provided, so I will demonstrate solving a simple algebraic equation as an example: Solv

1. مسئله: عبارت $$\frac{(y001)^4 \cdot (0003)^\alpha}{12000}$$ را با استفاده از خواص توان‌ها تجزیه و ساده کنیم. 2. ابتدا باید بدانیم که اعداد داخل پرانتز را به صورت عددی ساده کنیم:

1. The problem is to explain why the equation $y = x + 9$ is a linear equation. 2. A linear equation in two variables $x$ and $y$ can be written in the form $$y = mx + b$$ where $m

1. Stating the problem: Evaluate the expression $\frac{\sqrt{(32)(165)(37)}}{\sqrt{25}\,(46)(53)} - 5\sqrt{(34)(37)}$. 2. Formula used: Use the product rule for square roots $\sqrt

1. The problem asks for the equation of a straight line passing through the point $(1, 5)$ and parallel to the line $y = -2x + 3$. 2. Recall that parallel lines have the same slope