🧮 algebra

1. El problema es graficar la función $$y = e^{2x} - x^2 + 1$$ y hallar el valor del vértice. 2. Para encontrar el vértice de una función, primero calculamos la derivada para encon

1. **Problem Statement:** Rationalise the denominators of the following expressions and simplify where possible. 2. **Formula and Rule:** To rationalise a denominator containing a

1. **Problema:** Dada la función cuadrática $$f(x) = x^2 - 2x + 1$$, hallar las asíntotas (si existen) y el valor del vértice. 2. **Fórmula y reglas importantes:**

1. El problema es calcular el producto de las fracciones $$\frac{11}{4} \times \frac{3}{5} \times \left(-1 \frac{1}{3}\right)$$. 2. Primero, convertimos el número mixto $$-1 \frac{

1. **State the problem:** Simplify the expression $\sqrt{50} + \sqrt{78}$. 2. **Recall the formula and rules:** The square root of a product can be expressed as the product of squa

1. **Problem statement:** Find the remainder when $f(x) = 3x^3 - 14x^2 - 47x - 14$ is divided by $(x - 3)$.

1. El problema es entender con qué método se simplifica el numerador y denominador de una fracción compleja. 2. Una fracción compleja es una fracción donde el numerador, el denomin

1. Planteamos el problema: Simplificar completamente la expresión $$\left[ \frac{ \frac{2}{x-1} - \frac{3}{x+2}}{\frac{1}{x-1} + \frac{1}{x+2}} \right] \div \left( \frac{x^2 - 4}{x

1. **State the problem:** We have the function $$f(x) = \frac{x^3 - 4x^2 + 1}{x - 2}$$. We need to find its domain, simplify it if possible, determine the nature of the discontinui

## Problem Let $f(x)=\frac{x^3-4x^2+1}{x-2}$. Find the domain of $f$, simplify where possible, determine whether $x=2$ is removable or a vertical asymptote, and compute $\lim_{x\to

1. **Problem statement:** Given the function $f(x) = \frac{x^3 - 4x^2 + 1}{x - 2}$, find its domain, simplify the expression if possible, determine the nature of the discontinuity

### Problem Sara has 24 apples. She gives 7 apples to her friend and then buys 15 more apples. How many apples does she have now?

1. **State the problem:** Solve for $a$ in the equation $$a^3 + a^2 = 36.$$\n\n2. **Rewrite the equation:** Factor out the common term $a^2$ from the left side:\n$$a^3 + a^2 = a^2(

1. State the problem: Solve for $a$ in the equation $a^3 + a^2 = 36$. 2. Factor the left side (common factor $a^2$):

1. Vamos resolver a equação dada: $$\frac{x}{\sqrt{x^2 + 9}} + 2 \cdot \frac{8 - x}{\sqrt{(8 - x)^2 + 4}} = 0$$

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$ This is a c

1. State the problem: Find $\lim_{x\to 3}\dfrac{x^2-9}{x-3}$.\n\n2. Identify a removable discontinuity: Since $x^2-9=(x-3)(x+3)$, the fraction becomes simpler and the limit can be

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $$a$$. 2. **Rewrite the equation:** We want to find $$a$$ such that $$a^3 + a^2 - 36 = 0$$.

1. **State the problem:** We have counts of consumers shopping at different stores and combinations of stores. We want to find: - The percent of consumers who shopped at more than

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$ This is a c

1. **State the problem:** Solve for $a$ in the equation $$a^3 + a^2 = 36.$$\n\n2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$\n\n3. **Factor the