🧮 algebra

1. **Problem statement:** Determine if there exists a standard graded algebra $R$ such that $\dim R_1=4$ and $\dim R_3=3$. 2. **Recall definitions:** A standard graded algebra $R=\

1. **State the problem:** Solve the quadratic equation $$9x^2 + 16 = 24x$$ using the quadratic formula. 2. **Rewrite the equation in standard form:** Move all terms to one side:

1. مسئله: حل معادله $2x + 3 = 7$. 2. فرمول: برای حل معادله خطی، هدف این است که $x$ را تنها کنیم.

1. **State the problem:** Solve the equation $$4x^{-4} - 5x^{-2} + 1 = 0$$ for $x$. 2. **Rewrite the equation using substitution:** Let $y = x^{-2}$. Then $x^{-4} = (x^{-2})^2 = y^

1. مسئله: حل معادله $2x + 3 = 7$. 2. فرمول: برای حل معادله خطی، باید $x$ را تنها کنیم.

1. مسئله: تعیین اعضای مجموعه‌های داده شده است. 2. مجموعه اول: $\{2^{xy} \mid x,y \in \mathbb{N}, x + y = 5\}$

1. **State the problem:** Solve the equation $$x + \sqrt{11x + 44} = -4$$ for $x$. 2. **Isolate the square root term:** Move $x$ to the right side:

1. The problem is to solve the equation $$(x + 8)^2 = 39$$ for $x$. 2. We use the special quadratic equation rule: If $$x^2 = d$$, then $$x = \pm \sqrt{d}$$.

1. **State the problem:** Solve the equation $$\frac{5}{x-1} + \frac{1}{4-3x} = \frac{3}{6x-8}$$ for $x$. 2. **Rewrite the denominators:** Notice that $$4-3x = -(3x-4)$$ and $$6x-8

1. The problem is to understand the quadratic expression $x^2 + bx + c$ and how to work with it. 2. This is a quadratic polynomial where $b$ and $c$ are constants.

1. مسئله: حل معادله درجه دوم $7x^2 - 21x = 0$ به روش تجزیه. 2. فرمول و روش: برای حل معادله درجه دوم به روش تجزیه، ابتدا عامل مشترک را خارج می‌کنیم.

1. مسئله: حل معادله درجه دوم 7x^2 - 21x = 0 به روش تجزیه. 2. فرمول و روش: برای حل معادله درجه دوم به روش تجزیه، ابتدا عبارت را به صورت حاصل‌ضرب دو عبارت خطی بنویسید و سپس هر عامل ر

1. **Problem Statement:** A total sum of 700 is to be distributed as seven cash prizes such that each prize is 20 less than the previous one. We need to find the value of each priz

1. **State the problem:** Simplify or express the sum $a^4 + b^4$ in a useful form. 2. **Formula and rules:** There is no simple factorization over real numbers for $a^4 + b^4$ lik

1. The problem is to find the value of $x$ in the equation $$\frac{2x+3}{4} = 5.$$\n\n2. The formula used here is to solve linear equations by isolating the variable $x$.\n\n3. Fir

1. **Problem statement:** Simplify the expression $$\frac{5^{m+3} + 125 \cdot 5^{m-1}}{5^{m+4} \div 25}$$. 2. **Recall important rules:**

1. **State the problem:** Solve the equation $$\sqrt{3} + 2x = -x$$ for $x$. 2. **Rewrite the equation:** Move all terms involving $x$ to one side:

1. The problem is to analyze the summation $$\sum_{i=1}^n iL(2)^{x+2} = (n-2)^{x+2} A$$ with the condition $$n > 0$$. 2. First, understand the summation notation: $$\sum_{i=1}^n iL

1. **Stating the problem:** We want to verify or understand the formula for the sum $$\sum_{i=1}^{n+1} i(2^i) = n \cdot 2^{n+2} + 2$$. 2. **Recall the formula for the sum of the fo

1. **State the problem:** Solve the quadratic equation $2x^2 - 13x + 20 = 0$. 2. **Formula used:** The quadratic formula is given by

1. **Problem:** Find the value of $32 \equiv \, ? \pmod{12}$. 2. **Formula:** To find $a \equiv r \pmod{m}$, compute the remainder $r$ when $a$ is divided by $m$.