🧮 algebra

1. Planteamos el problema: Resolver la ecuación $$\frac{5^{x^2-1}}{25^{x+1}} = \frac{1}{125}$$. 2. Recordemos que $25 = 5^2$ y $125 = 5^3$, por lo que podemos reescribir la ecuació

1. Planteamos el problema: Encontrar las asíntotas vertical y oblicua de la función $$f(x)=\frac{3x^2-5x+1}{x+2}$$. 2. Asíntota vertical: Ocurre donde el denominador es cero y el n

1. **State the problem:** Simplify the expression $$\{[8x-5+(2x-3)]-[-4x+7-(x+2)]\}$$ and find the result. 2. **Rewrite the expression by removing brackets carefully:**

1. **Problema:** Graficar y analizar la función $y = x^2 - 6x + 8$. 2. **Fórmula y reglas:** La función es cuadrática y tiene la forma general $y = ax^2 + bx + c$.

1. The user asked to graph a function but did not specify which function to graph. 2. To graph a function, we need the explicit formula or expression for the function.

1. **State the problem:** Simplify the surds (square roots) of the numbers 12, 28, 32, 125, 300, 72, and 20. 2. **Formula and rules:** To simplify a surd $\sqrt{n}$, find the large

1. **Problem Statement:** Find the equation of the line $y = mx + c$ for graph (a) from Question ONE. 2. **Formula and Rules:**

1. Vấn đề: Tìm nghiệm chính xác của phương trình hoặc bài toán bạn cần giải. 2. Công thức và quy tắc: Để tìm nghiệm chính xác, ta cần biết dạng phương trình hoặc bài toán cụ thể để

1. **Nêu bài toán:** Doanh nghiệp sản xuất $x$ sản phẩm với $1 \leq x \leq 400$. Hàm doanh thu là

1. **State the problem:** Solve the equation $$\frac{3}{2}x - 2.5 = 5 + 0.25x$$. 2. **Write the formula and rules:** To solve linear equations, isolate the variable $x$ by moving a

1. The problem is to resolve the number 19, which means expressing 19 as a sum or product of other numbers or factors. 2. Since 19 is a prime number, it cannot be factored into sma

1. **State the problem:** Raj has a total of 19 units of money composed of $1 coins, $2 notes, and $5 notes. He has 9 coins and notes in total.

1. **Planteamiento del problema:** Queremos encontrar la ecuación vectorial de la recta $M$ que pasa por el punto $P=(7,2)$ y tiene como vector dirección $\vec{v} = (5, -6)$.

1. **Problem statement:** Solve the system of equations using the addition (elimination) method. 2. **Method:** The addition method involves adding or subtracting the equations to

1. **State the problem:** We need to fill in digits 1 to 9 in three logarithmic expressions to produce specified types of numbers: an integer, an irrational number, and a rational

1. **State the problem:** Find the equation of the line joining the points (4,1) and (7,3). 2. **Formula used:** The slope $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$

1. **State the problem:** Find the equation of the line passing through the points (4,1) and (7,3), and also the line passing through (-5,1) and (-3,-2). 2. **Formula used:** The s

1. **State the problem:** Simplify the expression $-(-526+-350)-150+74$. 2. **Rewrite the expression:** Note that $-(-a) = a$, so

1. **Problem:** Find the equation of the line joining the points (4,1) and (7,3). 2. **Formula:** The slope $m$ of the line through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

1. **State the problem:** We need to find the range of values of $a$ (where $a \neq 0$) such that the cubic equation $$2x^3 - 3ax^2 + a^3 - 2a = 0$$ has exactly one real root. 2. *

1. **Problem 2:** Find the gradient for each line and express the equation in the form $ax+by+c=0$. 2. **Problem 3:** Find the equation of the line through point $(2,5)$ with gradi