🧮 algebra

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

1. The problem is to find the equation of the line passing through the points $(-3, 4)$ and $(2, -1)$. 2. The formula to find the slope $m$ of a line passing through two points $(x

1. **State the problem:** We are given the function $$f(x) = -1 |x + 2| + 6$$ and asked to understand its graph and properties. 2. **Formula and explanation:** The function is an a

1. **نص المشكلة:** لدينا فصل به عدد من الطلاب. إذا جلس كل ٤ طلاب على طاولة، تبقى طاولات فارغة (أي لا يتم استخدام كل الطاولات). وإذا جلس كل ٣ طلاب على طاولة، تبقى ٣ طلاب بدون أماكن.

1. **State the problem:** We have a cubic function $f(x) = a x^3 - 2 x^2 + 4$ and know it has an inflection point at $x = \frac{1}{3}$. We need to find the value of $\frac{f(1)}{f(

1. **State the problem:** Find coefficients $a$, $b$, $c$, and $d$ for the cubic function $$y = ax^3 + bx^2 + cx + d$$ such that it has a critical point at $(2,0)$ and an inflectio

1. **Problem Statement:** You need to design a community park using algebraic expressions to represent costs, areas, and quantities, then simplify and evaluate these expressions to

1. Masla samajhna: Reciprocal ka matlab hota hai kisi number ka ulta ya inverse. 2. Formula: Agar kisi number ko $x$ kaha jaye, to uska reciprocal hota hai $\frac{1}{x}$.

1. **State the problem:** Solve the quadratic equation $6x^2 + 8x - 12 = 0$. 2. **Formula used:** The quadratic formula is given by

1. The problem asks whether a horizontal asymptote can touch or cross the x-axis. 2. A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ appr

1. The problem is to find the reciprocal function of a given function. 2. The reciprocal function of a function $f(x)$ is defined as $g(x) = \frac{1}{f(x)}$.

1. समस्या को समझें: "reciprocal function" का मतलब है एक ऐसा function जो किसी संख्या का उल्टा (reciprocal) देता है। 2. सूत्र: किसी भी संख्या $x$ का reciprocal होता है $\frac{1}{x}$,

1. **State the problem:** We need to sketch the graph of the function $$f(x) = \frac{1}{x^2 + x - 12}$$. 2. **Factor the denominator:** To understand the behavior of the function,

1. **State the problem:** We are given four variables $N_1$, $N_2$, $N_3$, and $V_4$ defined by algebraic expressions involving $f$, $g$, and $h$, which themselves depend on $\phi$

1. The problem is to sketch the graph of the function $y = 18$. 2. This is a constant function where the output value is always 18 regardless of the input $x$.

1. Diketahui sistem pertidaksamaan linear: \begin{cases}