🧮 algebra

1. Problem: Determine the domain of the functions \(f(x) = \frac{\arcsin(x+1)}{2x+1} - \sqrt{8^x - 2^{x+2}}\) and \(f(x) = e^{x^2 - 1} \cdot \frac{\sqrt{\arccos(x-1)}}{x+2}\). 2. F

1. The problem is to compare the ellipse equation $$\frac{x^2}{64} + \frac{y^2}{100} = 1$$ with the standard form of an ellipse centered at the origin: $$\frac{x^2}{b^2} + \frac{y^

1. The problem states the equation of an ellipse centered at the origin with the major axis on the y-axis is given by: $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b$$

1. **State the problem:** Solve the quadratic equations and analyze the given points for the parabola. 2. **First equation:** $D = 36 - 4 \cdot 9$

1. **Problem statement:** We have a polynomial $P(x)$ that leaves a remainder 7 when divided by $(x-2)$ and a remainder 1 when divided by $(x+1)$. We want to find the remainder whe

1. **State the problem:** We need to find the values of $x$ and $y$ given the matrix equation: $$\left(4\begin{bmatrix}1 & 2 & 0 \\ 2 & -1 & 3\end{bmatrix} - 2 \begin{bmatrix}1 & 3

1. Problem: Verify if the statement sgn(−x) = −sgn(x) for all real numbers x is true. Step 1: Recall the definition of the sign function:

1. **State the problem:** We have the parabola given by $$(y+4)^2 = \frac{1}{2}(x-2)$$ and we want to find the equation of this parabola after rotating it by an angle of 60° anticl

1. **Problem Statement:** Find the least common multiple (LCM) of the given numbers and algebraic expressions. 2. **Recall the formula and rules:**

1. **State the problem:** Fully factorise the quadratic expression $n^2 + 9n + 20$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers th

1. **Problem statement:** Simplify the given rational expression and verify the partial fraction decomposition: $$\frac{4}{(2x-1)(x^2+4)} = \frac{16/17}{2x-1} + \frac{-8/17 x - 4/1

1. **State the problem:** Fully factorise the quadratic expression $x^2 - x - 42$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers tha

1. **Problem Statement:** We are given a square with side length expressed as $4x + 3$. We want to understand properties related to this square, such as its perimeter and area. 2.

1. The problem asks us to find the value of $x$ in the equation $$\frac{2x+3}{4} = 5.$$\n\n2. The formula used here is to solve linear equations by isolating the variable $x$. We d

1. **State the problem:** We need to solve the algebraic expression or equation given by the user. Since no specific problem was provided, please provide the exact problem for deta

1. **State the problem:** We need to solve the equation or analyze the function given (please provide the specific problem to proceed). 2. **Formula and rules:** Depending on the p

1. **Stating the problem:** We are given a triangle divided into three parts with numbers in the bottom row and a missing top number represented by a question mark (?). We need to

1. **State the problem:** We analyze the function $$f(x) = \frac{(3x + 2)^2}{(x - 2)^2}$$ to find: (a) Vertical asymptotes

1. Das Problem lautet: Gegeben ist die Funktion $g(x) = x^3 - 2x^2 + x$. Wir sollen den Funktionswert an der Stelle $x=4$ berechnen, also $g(4)$. 2. Die Formel zur Berechnung des F

1. The problem is to find the missing coordinate of point P given as $P(4|?)$. 2. To find the missing coordinate, we need more information such as an equation, a line, or a curve t

1. Wir haben die Funktion gegeben: $g(x) = x^3 - 2x^2 + x$. 2. Ziel ist es, die Funktion zu analysieren, z.B. Nullstellen, Extremstellen und Verhalten zu bestimmen.