🧮 algebra

1. **Problem 3a:** Find the centre of the circle $C$ with equation $$x^2 + y^2 - 4x + 10y = k.$$ 2. **Step 1:** Rewrite the equation by completing the square for $x$ and $y$.

1. مسئله: ساده‌سازی عبارت‌های داده شده. 2. فرمول‌ها و قوانین مهم:

1. **Problem 25:** Which animal is represented by the point farthest from 0 on the number line? 2. **Step 1:** Identify the absolute values of each animal's location (distance from

1. Problemet er at differentiere funktionen $$f(x) = \ln(x^2 + 1) + 2x$$. 2. Vi bruger reglen for differentiering af en sum: $$\frac{d}{dx}[u + v] = \frac{du}{dx} + \frac{dv}{dx}$$

1. Problemet er at finde udtryk for \( \ln(x^2+1) \).\n\n2. Vi ved, at \( \ln(a) \) er den naturlige logaritme af \( a \), og den er defineret for \( a > 0 \). Her er argumentet \(

1. **Stating the problem:** A knight must arrive at exactly 17h00. Traveling at 15 km/h makes him arrive 1 hour early, and at 10 km/h makes him arrive 1 hour late. We need to find:

1. Énoncé du problème : Résoudre l'équation $$\frac{2}{2x+1} - \frac{4}{x} = 0$$ et déterminer les valeurs interdites. 2. Valeurs interdites : Ce sont les valeurs de $x$ qui renden

1. **State the problem:** We need to find how many paper chains used more than $2 \frac{3}{4}$ containers but less than 4 containers of glitter. 2. **Identify the relevant data:**

1. Vi har funktionen $f(x) = \frac{2x}{x^2 + 1}$. 2. Opgaven er at finde en stamfunktion $F(x)$ til $f(x)$, altså en funktion hvor $F'(x) = f(x)$.

1. **State the problem:** We need to graph the linear function $$y = 4x + 1$$ on a coordinate plane. 2. **Formula and explanation:** This is a linear equation in slope-intercept fo

1. **Stating the problem:** We need to find the sum of cubes of integers from 50 to 100, i.e., calculate $$\sum_{k=50}^{100} k^3$$. 2. **Formula used:** The sum of cubes from 1 to

1. **State the problem:** Simplify or factor the quadratic expression $x^2 + 7x + 12$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers

1. The problem is to calculate $2$ raised to the power of $6$, which means multiplying $2$ by itself $6$ times. 2. The formula for exponentiation is $a^n = a \times a \times \cdots

1. Problem: Calculate $2$ raised to the power of $7$. 2. Formula: The expression $a^b$ means multiplying $a$ by itself $b$ times.

1. The problem is to evaluate the expression $3 - 1_2$. 2. The notation $1_2$ means the number 1 in base 2 (binary).

1. Problem: Czy $-\frac{3}{3}$ to jest $-3$ całe? 2. Wyjaśnienie: $-\frac{3}{3}$ oznacza ułamek, gdzie licznik to $-3$, a mianownik to $3$.

1. **Problem statement:** Two pipes can fill a tank together in 6 hours. The larger pipe works twice as fast as the smaller pipe. We need to find how long each pipe would take to f

1. **State the problem:** Simplify the expression $$\frac{4y^3 - 8y^2 + 7y - 14}{-y^2 - 5y + 14}$$. 2. **Factor numerator and denominator:**

1. **State the problem:** Simplify the expression $$\frac{2x^3 - 6x^2 + 5x - 15}{9 - x^2}$$. 2. **Identify the formula and rules:** We will factor both numerator and denominator wh

1. **Stating the problem:** We are given multiple systems of linear inequalities and asked to analyze and understand the solution regions represented by shaded triangular areas on

1. The problem is to solve the equation $2x = $ (incomplete equation). 2. To solve for $x$, we need a complete equation, for example, $2x = 10$.