🧮 algebra

1. **Question 1: Calculate the exact value of** $$Q = \frac{(\sin 2x + b)(2 \sin x - 1)}{a^2 - 4 \tan x}$$

1. نبدأ بتعريف الدوال: الدالة $g(x) = x^3 + 6x - 4$ وهي كثيرة حدود تكعيبية.

1. **State the problem:** A student must attend at least 160 days but not more than 200 days to meet attendance requirements. 2. **Write the inequality:** Let $d$ be the number of

1. Énonçons le problème : Calculons $C = 3\sqrt{75} - 2\sqrt{139} + 3\sqrt{182} - \sqrt{432}$ en simplifiant chaque terme radical. 2. Simplifions radicaux un par un.

1. **Find the 10th term of the arithmetic sequence where $a_1=5$ and $d=3$.** The $n$th term of an arithmetic sequence is given by:

1. Simplifions les nombres donnés. a = \frac{\sqrt{18} \times \sqrt[3]{256} \times \sqrt[4]{64}}{\sqrt[3]{1024} \times \sqrt[6]{64} \times 10^{6}}

1. **Stating the problem:** Let's understand arithmetic sequences and series step-by-step. 2. **Arithmetic Sequence:** It is a list of numbers with a constant difference between co

1. Diberikan limit $$\lim_{x \to 1} \frac{x^4 - 1}{x^2 + 5x - 6}$$. 2. Faktorkan pembilang: $$x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)$$.

1. The problem is to convert the repeating decimal $1.8\overline{8}$ into a fraction. 2. Let $x = 1.8\overline{8}$ represent the number.

1. Pernyataan masalah: Tentukan pernyataan yang bersesuaian dengan $$\lim_{x \to 4} \frac{2-\sqrt{x}}{4-x}$$. 2. Perhatikan bahwa jika kita substitusi langsung $x=4$, maka pembilan

1. Classify the following statements as true or false: (a) $-2 \in \mathbb{N}$: False, natural numbers are positive integers starting from 1.

1. Stated problem: Calculate $9 \frac{1}{6} \div \left(2 \frac{5}{6} - 1 \frac{1}{2}\right)$. 2. Convert mixed numbers to improper fractions:

1. **State the problem:** We are analyzing the expression $$\frac{2 - n}{\sqrt{n^2 + 3}}$$ and explaining why this expression is limited in terms of its domain and behavior. 2. **U

1. **State the problem:** Simplify the expression $$\frac{5}{6} \div 2 \frac{1}{3} + \frac{1}{2}$$. 2. **Convert mixed number to improper fraction:**

1. **Problem 1:** Find $x$ given that $|P| = -10$, where $$P = \begin{bmatrix} x+3 & x+2 \\ x+1 & x-1 \end{bmatrix}.$$ The determinant of $P$ is $$|P| = (x+3)(x-1) - (x+2)(x+1).$$

1. We are given the equation modeling the miles remaining to be paved after $x$ hours: $$y = 200 - 50x$$

1. The problem is to solve a quadratic equation generally expressed as $$ax^2 + bx + c = 0$$ where $a \neq 0$. 2. To solve it, we use the quadratic formula:

1. **State the problem:** Simplify the expression $4(5a + 3b) + 3(2a - b)$.\n\n2. **Distribute the coefficients:** Apply the distributive property by multiplying each term inside t

1. The problem asks to find the intersection $P \cap Q$ where: - $P$ is the set of prime factors of 210.

1. **State the problem:** Solve for $a$ given the equation $a^{\frac{1}{8}} = \sqrt{3}$.\n\n2. **Rewrite the equation:** Note that $\sqrt{3} = 3^{\frac{1}{2}}$. So the equation bec

1. **Énoncé du problème** : Montrer que $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \cdots + \frac{1}{\sqrt{99}+\sqrt{100}} = 9.$$\n