🧮 algebra

1. **State the problem:** We have an arithmetic sequence $2, 4, 6, 8, \ldots$ and want to find how many terms sum to 600. 2. **Recall the formula for the sum of an arithmetic seque

1. **Énoncé du problème :** Montrer plusieurs propriétés sur les sous-groupes $H$, $K$ de $G$ avec $H$ normal dans $G$. 2. **Définition et rappel :** $HK = \{hk \mid h \in H, k \in

1. **State the problem:** Simplify the expression $$\frac{x - 2}{x^2 - 16} = x(x + 4)(x - 4) - x(x + 4)(x - 4) \over x^2 - 4x$$. 2. **Rewrite the expression:** Notice the right sid

1. Zadatak: Riješiti matričnu jednadžbu $$X \cdot A - 2B = X$$ gdje su $$A = \begin{bmatrix}2 & -1 \\ 5 & 3 \end{bmatrix}, \quad B = \begin{bmatrix}0 & -1 \\ 3 & 5 \end{bmatrix}.$$

1. Calculer et simplifier : - $$\sqrt{12} \times \sqrt{3} = \sqrt{12 \times 3} = \sqrt{36} = 6$$

1. Planteamos el problema: Dada la función $$f(x) = \log_4(x+2) + \log_4(x-2) - \log_4 5,$$ queremos encontrar para qué valores de $x$ se cumple que $f(x) < 0$. 2. Recordemos que e

1. **Problem statement:** Find a rational function $f$ defined on its maximum domain that has a vertical asymptote at $x=7$. 2. **Recall:** A vertical asymptote occurs where the de

1. **Stating the problem:** We need to show that the determinant of the matrix $$\begin{vmatrix}-a^2 & 2ab & ac \\ ba & -b^2 & bc \\ ca & cb & -c^2\end{vmatrix} = 4a^2b^2c^2.$$

1. **Énoncé du problème :** Déterminer l'ensemble de définition de la fonction $$f(x) = \frac{x - 1}{|3x - 1|} - 4$$. 2. **Rappel des règles importantes :**

1. The problem is to understand the number 3.91938417383707E-28. 2. This number is written in scientific notation, where "E-28" means \times 10^{-28}.

1. The problem is to understand and express the number 3.91938417383707E+28 in a more readable form. 2. The notation 3.91938417383707E+28 means $3.91938417383707 \times 10^{28}$.

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + y + 2z = 9 \\ 2x + 4y - 3z = 1 \\ 3x + 6y - 5z = 0 \end{cases}$$

1. The user requests to change the value in an equation and receive a PDF file with the changed value. 2. However, no specific equation or value to change was provided in the messa

1. The problem is to solve the equation $5x - 5 = -3$ for $x$. 2. The formula used here is to isolate $x$ by performing inverse operations step-by-step.

1. **Stating the problem:** Calculate the value of $A$ given the expression $A = 4 \times -5$ and $x = -3$. 2. **Understanding the expression:** The expression $A = 4 \times -5$ me

1. **Problem Statement:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula Used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the quad

1. Problema este să calculăm limita funcției $$\lim_{x \to 2} \frac{3x^2 - 5x + 1}{4x - 2}$$. 2. Formula pentru limită este evaluarea directă a expresiei dacă este posibil.

1. The problem is to simplify or verify the expression $5\sqrt{6}$ and compare it to the textbook answer $\frac{5\sqrt{6}}{2}$. 2. The expression $5\sqrt{6}$ means 5 times the squa

1. **State the problem:** Solve the equation $$\frac{p}{2} + \frac{p}{3} = 10$$ for $p$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to c

1. **Problem 1: Find the equation of a straight line** Given two points $A(2,3)$ and $B(5,11)$, find the equation of the line passing through these points.

1. **Problem 1: Find the equation of a straight line** Given two points on a line, $A(2,3)$ and $B(5,11)$, find the equation of the line passing through these points.