🧮 algebra

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 $$\lim_{x \to 3} \frac{x^2 + 5x + 1}{x^3 + 27}$$.

1. **State the problem:** We need to find the value of the blank rectangle in the equation $$\frac{2}{5} + \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \text{[blank rectangle]} \times

1. El problema nos presenta la ecuación $$\frac{7}{12} = [\quad] \times \frac{1}{12}$$ y nos pide completar el espacio en blanco. 2. Para resolver, recordemos que multiplicar por $

1. **State the problem:** Albert had 16.10 less than Peter. After Peter gave 2.50 to Albert, Peter had 4 times as much money as Albert. 2. **Define variables:** Let $P$ be the amou

1. **State the problem:** We need to find the square of the fraction $\frac{3}{2}$. 2. **Formula:** To square a fraction, square the numerator and the denominator separately:

1. Simplify each expression or ratio as given. **a)** Simplify $8 \times 8 \times 8 \times 8 \times 8 \times 8$.

1. **State the problem:** Solve the quadratic equation $-3x^2 + 2x + 1 = 0$. 2. **Formula used:** The quadratic formula is given by