🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{2x - 6 + (x^2 + 4x + 3)}{x^2 + 2x - 15} + \frac{1}{x + 5}$$

1. Planteamos el problema: Simplificar la expresión $$\frac{2x - 6}{1} + \frac{x^2 + 4x + 3}{x^2 + 2x - 15} + \frac{1}{x + 5}$$. 2. Factorizamos los polinomios donde sea posible pa

1. El problema es simplificar la expresión o resolver la ecuación dada (no especificada en el mensaje). 2. Para simplificar o resolver, se aplican reglas básicas de álgebra como co

1. **Problem statement:** Lisa has the digits 2, 0, 2, and 5 and wants to form the largest possible number using these digits. 2. **Formula and rules:** To form the largest number

1. Planteamos el problema: Resolver la ecuación \( \log_5 (2x - 3) = 3 \).\n\n2. Recordemos que \( \log_a b = c \) significa que \( a^c = b \).\n\n3. Aplicamos esta definición para

1. **State the problem:** We need to analyze the rational function $$f(x) = \frac{2x^3 + 3x^2 - 3x - 2}{2x^3 - 7x^2 + 2x + 3}$$ including simplification, domain, asymptotes, and be

1. **Planteamiento del problema:** Calcular las asíntotas verticales y horizontales de la función $$f(x) = \frac{2x^3 + 3x^2 - 3x - 2}{2x^3 - 7x^2 + 2x + 3}$$, clasificar las disco

1. **State the problem:** Solve the equation $ (x+3)^2 - (x^2 + 6x) = 5 $ for $x$. 2. **Recall formulas and rules:**

1. El problema es resolver una ecuación lineal, que es una ecuación de primer grado en la variable $x$. 2. La forma general de una ecuación lineal es $ax + b = 0$, donde $a$ y $b$

1. **State the problem:** Simplify the expression $a^3 - a^2 x + a x^2$. 2. **Identify common factors:** Notice that each term contains a factor of $a$.

1. Planteamos el problema: sumar las fracciones algebraicas $$\frac{1}{2x - 6} + \frac{x^2 + 4x + 3}{x^2 + 2x - 15} + \frac{1}{x + 5}$$. 2. Identificamos los denominadores y factor

1. Planteamos el problema: Resolver la ecuación $$25(x + 2)^2 = (x - 7)^2 - 81$$. 2. Usamos la fórmula de expansión de binomios cuadrados: $$(a+b)^2 = a^2 + 2ab + b^2$$.

1. **Problem Statement:** Add the polynomials $2x^2 - 5xy + y^3$ and $4x + y$. 2. **Formula and Rules:** To add polynomials, combine like terms. Like terms have the same variables

1. **State the problem:** Solve the equation $\frac{2x+4}{3} = 5$ for $x$. 2. **Formula and rules:** To solve equations involving fractions, multiply both sides by the denominator

1. The user requested a response with no numbers, only expressions or variables. 2. This means all mathematical expressions should be left in symbolic form without substituting any

1. The problem is to find the answer without multiplying by 10000. 2. If you have a number multiplied by 10000, for example $x \times 10000$, to get the answer without 10000, you s

1. **State the problem:** We have a cuboid with volume $2$ m³ and three sides: $5 - \sqrt{3}$ cm, $2 + \sqrt{3}$ cm, and $x$ cm (the missing side). We need to find $x$ in simplifie

1. **Problem Statement:** Simplify the expressions: d) $\frac{3 - \sqrt{2}}{6 - \sqrt{5}}$

1. **Problem d:** Simplify the expression $$\frac{3 - \sqrt{2}}{6 - \sqrt{5}}$$. 2. **Step 1:** Rationalize the denominator by multiplying numerator and denominator by the conjugat

1. **State the problem:** We have three lamps A, B, and C flashing at intervals of 20, 45, and 120 seconds respectively. They start flashing simultaneously at time zero. We want to

1. **Problem Statement:** Rationalise the denominators of the following expressions and simplify where possible: a) $\frac{1}{1+\sqrt{2}}$