🧮 algebra

1. **State the problem:** We are given the sum of the first $n$ terms of an arithmetic progression (AP) as $S_n = 3n^2 + 5n$. We need to find the $n$th term, $a_n$. 2. **Recall the

1. **Problem Statement:** Solve the equation $2x + 3 = 11$ for $x$. 2. **Formula and Rules:** To solve a linear equation, isolate the variable on one side by performing inverse ope

1. **Problem statement:** Prove that $$\log_2 \left[ \log_2 \left\{ \log_3 \left[ \log_3 27^3 \right] \right\} \right] = 0.$$\n\n2. **Recall the logarithm rules:**\n- $\log_a (b^c)

1. The problem is to solve or simplify an algebraic expression or equation using a method other than the one currently known or used. 2. Common alternative methods include substitu

1. **State the problem:** Make $x$ the subject of the equation $$y = \sqrt[3]{\frac{6 + 5x}{x + 4}}.$$ 2. **Rewrite the equation:** Cube both sides to eliminate the cube root:

1. **State the problem:** Show that $\frac{2\sqrt{3}}{\sqrt{3} - 1}$ can be written in the form $a + \sqrt{a}$ where $a$ is an integer. 2. **Rationalize the denominator:** Multiply

1. Muammo: Tenglama $\sqrt{x^4} + 5x = -3x$ ildizlari yig'indisini toping. 2. Avvalo, $\sqrt{x^4} = |x^2| = x^2$ (chunki $x^2$ har doim manfiy emas).

1. Muammo: Tenglama $\sqrt{x^4} + 5x = -3x$ ildizlari yig'indisini topish. 2. Avvalo, $\sqrt{x^4} = |x^2| = x^2$ (chunki $x^2$ har doim manfiy emas).

1. Factorize the polynomial $21pq + 14q - 28qr$. 2. Identify the greatest common factor (GCF) of all terms.

1. **Problem Statement:** We have a triangular plot with teams Alpha, Beta, and Charlie at its corners. The area of the triangle is 37 sq units. Charlie lies on the positive X-axis

1. **State the problem:** Solve the equation $$x(y-z)p + y(z-x)q = \frac{x-y}{xy}$$ for variables $p$ and $q$. 2. **Understand the equation:** This is a linear equation in terms of

1. The problem is to simplify the expression $(\times)(Cx \quad Dx)$. 2. First, clarify the meaning of the symbols: if $C$ and $D$ are constants or variables and $x$ is a variable,

1. Дано уравнение: $$x^2 + 5x - 2(x^2 + 3) = -x^2 - 3x + 11$$ 2. Раскроем скобки слева: $$x^2 + 5x - 2x^2 - 6 = -x^2 - 3x + 11$$

1. **Énoncé du problème 1** : Trouver trois entiers $a$, $b$, $c$ tels que $$\frac{5}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$ pour $n=3$, $n=4$ et $n=9$. 2. **Méthode** : On

1. **Problem:** Quadrilateral WXYZ is mapped onto W'X'Y'Z' by transformation matrix $$P=\begin{pmatrix}1 & 3 \\ -3 & 1\end{pmatrix}$$. Given coordinates of W'X'Y'Z' as $$W'(3,-2),

1. المشكلة: حل معادلة من الدرجة الثانية. 2. الصيغة العامة للمعادلة التربيعية هي $$ax^2 + bx + c = 0$$ حيث $a \neq 0$.

1. The problem is to understand and analyze the function $g(x) = (s+1)e^x$. 2. The function $g(x)$ is a product of a constant term $(s+1)$ and the exponential function $e^x$.

1. Bài toán yêu cầu tìm tất cả các giá trị của tham số $m$ sao cho hàm số $f(x) = x^3 + m x^2 - 3x - m + 2$ luôn dương với mọi $x$.\n\n2. Để hàm số luôn dương, ta cần $f(x) > 0$ vớ

1. The problem is to verify or find the value of a given expression that results in 2.666. 2. Since the user states the answer is 2.666, let's consider it as a decimal approximatio

1. **Stating the problem:** We are given examples of rational numbers: $-\frac{5.6}{9}$, $0.25$, and $x \times 8$. We need to calculate the value of $x$ such that $x \times 8$ is a

1. **Stating the problem:** We want to understand and possibly solve the equation $v_{tran} = w \log_2(1 + p_{gt}^n)$.