🧮 algebra

1. **State the problem:** Given that $m+n=3$, find the value of the expression $$2m^2 + 4mn + 2n^2 - 6.$$ 2. **Rewrite and simplify the expression:** Notice we can group the terms

1. مسئله: معادله $x^2 + mx + 2 = 0$ دارای دو ریشه حقیقی متمایز و هر دو ریشه منفی است. 2. برای معادله درجه دوم $x^2 + mx + 2 = 0$ ریشه ها را با $x_1$ و $x_2$ فرض کنیم.

1. **State the problem:** We are given positive integers $a$ and $b$ such that $$a^2 + 2ab - 3b^2 - 41 = 0$$ and we need to find the value of $$a^2 + b^2$$. 2. **Rewrite the given

1. The problem states we have a polynomial $$P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ with non-negative integer coefficients $a_i$. 2. Given:

1. Given are several equations involving variables $P$ and $Q$: $P = -4Q + 40000$

1. Problem: Find the negative integer solutions to the inequality $x > -4$. The integers greater than $-4$ are $-3, -2, -1, 0, 1, \ldots$, but the problem asks for negative integer

1. **State the problem:** We are given a function $$f(m,n) = \prod_{k=m}^n \left(1 - \frac{1}{k}\right)$$

1. Nelygybės $x > -4$ sveikieji neigiami sprendiniai yra tie skaičiai, kurie yra maži už $-4$ arba mažesni, bet neigiami ir sveikieji. Sveikųjų neigiamų skaičių intervalas nuo $-3$

1. **State the problem:** Given positive real numbers $a$, $b$, and $c$ satisfying $$\frac{a}{a+b} = \frac{a+b}{a+b+c} = \frac{c}{b+c}$$

1. First, we state the problem: Given integers $m$ and $n$ with $1 < m \le n$, define $$f(m, n) = \left(1 - \frac{1}{m}\right) \times \left(1 - \frac{1}{m + 1}\right) \times \cdots

1. Masalah yang diberikan ialah ketaksamaan: $$-5x < 15$$. 2. Untuk menyelesaikan ketaksamaan ini, kita perlu membahagikan kedua-dua pihak dengan $$-5$$.

1. Stating the problem: Simplify the expression $\frac{7}{8} + \frac{4}{1} \div 4 + \frac{15}{16}$. 2. First, evaluate the division part: $\frac{4}{1} \div 4 = \frac{4}{1} \times \

1. The problem asks to solve the inequalities involving absolute values: (i) \(|x| < 2\)

1. **State the problem:** We have positive real numbers $a$, $b$, $c$ such that $$\frac{a}{a+b} = \frac{a+b}{a+b+c} = \frac{c}{b+c}$$

1. **State the problem:** We are given a sequence of values $a_1 = 16944930$, $a_2 = 36736961$, $a_3 = 42442472$, $a_4 = 418656672$, $a_5 = 435042998$. We want to find the common d

1. **State the problem:** We have two quadratic equations: - Equation 1: $2x^2 - mx + 8 = 0$ with roots $\alpha$ and $\beta$.

1. We are given the system of equations: $$xy=12$$

1. The problem is to solve the simultaneous equations. Since the specific equations are not provided, I will explain the general method. 2. Suppose the simultaneous equations are:

1. The problem requires the lumber quantity to be an integer value. 2. Let's assume the previous question involved calculating the amount of lumber needed as a number, possibly a d

1. **State the problem:** You have 3 lumber units to spend. Each lumber can buy 160000 HP directly.

1. The problem is to verify whether the correct fraction is $\frac{135}{574}$ or $\frac{135}{287}$.\n\n2. Check if $\frac{135}{287}$ can be simplified or if it matches any known so