🧮 algebra

1. **State the problem:** We are given the function $$f(x) = \frac{3x^2 + 16}{(1 - 3x)(2 + x)^2}$$ and the partial fraction decomposition form:

1. **State the problem:** Given that $3^x = 16$, find the value of the sum $|x-2| + |x-3|$. 2. **Find $x$:** We start by solving for $x$ in the equation $3^x = 16$.

1. The problem is to analyze the function $f(x) = x^3 - 12x^2 + 36x$ and determine for which integer values of $x$ from 0 to 6 the function value is at least 5. 2. The function is

1. Énoncé du problème : Trouver le coefficient directeur de la tangente à la courbe de la fonction $f(x) = \frac{x^2}{x+1}$ au point d'abscisse $x=1$. 2. Rappel de la formule : Le

1. **Problem Statement:** Solve the linear equation $3x + 5 = 20$ for $x$. 2. **Formula and Rules:** A linear equation in one variable has the form $ax + b = c$, where $a$, $b$, an

1. **Problem statement:** Given the inequalities $4 < y - x < 9$ and $10 < x^2 + y^2 < 45$, and the functions $B = \frac{x+1}{2}$ and $A = \sqrt{\frac{x^2 + y^2}{y - x}}$, we need

1. **State the problem:** We need to evaluate the expression $x - \sqrt{3x}$ given that $x - \sqrt{\frac{12}{x}} = 5$. 2. **Analyze the given equation:**

1. **Stating the problem:** We need to determine which of the given sequences is not an arithmetic progression (A.P.). 2. **Recall the definition of A.P.:** A sequence is an arithm

1. **Problem statement:** Find the value of $x$ such that $2x$, $x+10$, and $3x+2$ are three consecutive terms of an arithmetic progression (A.P.). 2. **Formula and rule:** In an A

1. **Problem statement:** We want to find the integer $a$ in the fourth term of the binomial expansion of $(1.02)^6$, given the first term is 1 and the fourth term is $(0.000008)a$

1. **State the problem:** Simplify the expression $$\sqrt[3]{8x} \cdot \sqrt[3]{8x^2}$$. 2. **Recall the property of cube roots:** For any real numbers $a$ and $b$, $$\sqrt[3]{a} \

1. The problem involves inverse proportionality, where two variables multiply to a constant. For example, if $g$ and $h$ are inversely proportional, then $g = \frac{k}{h}$ for some

1. **Stating the problem:** You get 20 cents for each correct answer and pay 10 cents for each wrong answer. After 30 questions, your profit is 1.80 dollars. 2. **Define variables:

1. **State the problem:** We are given a geometric progression (GP) with the first term $a=3$ and common ratio $r=3$. We need to find which term of this series is equal to 243. 2.

1. **Stating the problem:** We are given a set of points $(x, y)$ and asked to find the equation of the trendline that best fits the data. 2. **Analyzing the data:** The $x$ values

1. **Stating the problem:** We have a water tank with total capacity $y$ gallons. Initially, it contains $c$ gallons of water. Water flows into the tank at a rate of $m$ gallons pe

1. **Problem:** Given that the matrix below is singular, find the possible values of $x$ for the matrix $$\begin{bmatrix} x & 1 \\ 2 & 2 \end{bmatrix}$$ 2. **Formula and rule:** A

1. **Stating the problem:** We are given a set of data points $(x, y)$ and want to find an exponential model of the form $$y = a \cdot e^{bx}$$ that best fits the data. 2. **Formul

1. The problem is to find the equation of the exponential trendline for the given data points using Excel. 2. The general form of an exponential function is $$y = a e^{bx}$$ where

1. **Problem statement:** Given a cubic equation $$x^3 + ax^2 + bx + c = 0$$ with roots $$\alpha, \beta, \gamma \in \mathbb{C}$$ such that $$\alpha + \beta, \beta + \gamma, \gamma

1. **Stating the problem:** We are given two functions: $f(x) = x^2 + 2x + 5$ and $f(g(x)) = x^2 + 4$. We need to find the function $g(x)$.\n\n2. **Understanding the problem:** The