🧮 algebra

1. **Stating the problem:** We have four numbers: $1.5 \times 10^4$, $4.8 \times 10^6$, $3.2 \times 10^2$, and $7.5 \times 10^3$.

1. **Stating the problem:** We have four numbers in standard form: $1.5 \times 10^4$, $4.8 \times 10^6$, $3.2 \times 10^2$, and $7.5 \times 10^3$.

1. **State the problem:** Find the slant (oblique) asymptote of the function $$h(x) = \frac{x^2 - 9}{x}$$. 2. **Recall the rule for slant asymptotes:** A slant asymptote occurs whe

1. **State the problem:** Find the zeros of the function $$f(x) = \frac{x^2 + 14x + 49}{x^2 + 1}$$. 2. **Recall the rule for zeros of a rational function:** The zeros occur where t

1. **Problem statement:** Given functions $f(x) = \frac{x+1}{x-3}$ and $g(x) = x + 3$, find the composition $(f \circ g)(x)$ and determine its domain. 2. **Formula and rules:** The

1. **State the problem:** We are given a rectangle BCDE with side BC labeled as $4t - 17$ and side DE labeled as $6t - 29$. We need to find the length of side DE. 2. **Recall prope

1. The problem asks for the 6th term of a geometric sequence with the first few terms 1, 5, 25, ... 2. A geometric sequence has terms where each term is found by multiplying the pr

1. **State the problem:** Solve the quadratic equation $$3x^2 - 5x + 2 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equ

1. **State the problem:** Given that $a^3 + b^3 = 217$ and $a + b = 7$, find the value of $ab$. 2. **Recall the formula:** The sum of cubes can be expressed as

1. State the problem: If $a^3+b^3=217$ and $a+b=7$, find $ab$. 2. Use the identity for sum of cubes:

1. **State the problem** We have $f(x)=\frac{3}{4}x+10$ and $g(x)=x^2-3$.

1. State the problem: Factor the quadratic $x^2+5x+6$. 2. Use the factoring rule: find two numbers $a$ and $b$ such that $ab=6$ and $a+b=5$.

1. State the problem. We are given $a-b=2$ and $b-c=2$. Find the value of

1. State the problem: Solve for $a$ in $a^3+a^2=36$.\n 2. Write the equation and factor it: $$a^3+a^2=36$$\n

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. Planteamos el problema: Tenemos los costos de importación dados por $$ax^3 - bx^2$$ y la ganancia dada por $$2ax^5 - 2bx^4$$, donde $a$ y $b$ son constantes. 2. Para factorizar

1. Planteamos el problema: Se nos da la función de costos de importación para una cantidad $x$ de productos como $$ax^3 - bx^2$$ donde $a$ y $b$ son constantes. 2. Esta expresión e

1. **Stating the problem:** Given that $abc=1$, calculate the expression $$\frac{a}{ab+a+1} + \frac{b}{bc+b+1} + \frac{c}{ac+c+1}.$$\n\n2. **Rewrite each denominator using the cond

1. **Planteamiento del problema:** Calcular para cada función cuadrática las raíces, el vértice, el rango, la ordenada al origen y graficar la parábola.

1. **Problema:** Resolver la función cuadrática $2x^2 + x + 1 = 0$. 2. **Fórmula para raíces:** Usamos la fórmula cuadrática $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ donde $a=2$,

1. **Planteamiento del problema:** Queremos calcular el límite $$\lim_{x \to 3} \frac{x^2 + 5x + 1}{x^3 + 27}$$.