🧮 algebra

1. **State the problem:** Given the arithmetic sequence terms $a_2, a_6, a_{10}, a_{14}$ and the equation $$3(a_2 + a_6) + 2(a_6 + a_{10} + a_{14}) = 24,$$ find the sum of the firs

1. **Stating the problem:** We have a sequence $\{a_n\}$ with the first term $a_1 = 2$ and the recurrence relation $$a_{n+1} = \frac{a_n}{2} + \frac{1}{2n - 1}.$$ We need to find $

1. **State the problem:** Simplify the expression $$2\sqrt{3} - 5 - \frac{11}{\sqrt{12} - 1}$$. 2. **Recall the formula and rule:** To simplify expressions with radicals in the den

1. Let's start by stating the problem: Understanding every type of algebraic fraction. 2. An algebraic fraction is a fraction where the numerator and/or denominator are algebraic e

1. **State the problem:** We are given the system of equations: $$5x + 14y = 45$$

1. **State the problem:** We need to find which system of linear equations corresponds to the two lines shown on the graph. 2. **Analyze the graph:**

1. **State the problem:** Expand and simplify the expression $t(7t - 2) + 3t(4t - 5)$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the factor

1. **State the problem:** Expand and simplify the expression $x(4x - 2) + x(5x + 6)$. 2. **Use the distributive property:** Multiply $x$ by each term inside the parentheses.

1. **State the problem:** Match each function description to the correct graph type. 2. **Analyze each function:**

1. The problem is to understand and analyze the given equations: $$y = 1.165 x \cdot 1.023^t$$

1. **State the problem:** Find the domain of the function $$f(x) = \sqrt{x^2 - 9}$$. 2. **Recall the domain rule for square roots:** The expression inside the square root must be g

1. **State the problem:** We are given that the speed $v$ of a vehicle is directly proportional to the power $p$ and the square of the radius $r$ of its wheels. Given $v=40$ m/s wh

1. The problem is to find the value of $|(2+2i)+/2|$. 2. First, clarify the expression. It seems to be $\left|\frac{2+2i}{2}\right|$, which means the magnitude of the complex numbe

1. **State the problem:** Find the domain of the function $$f(x) = \sqrt[3]{x+3}$$. 2. **Recall the domain rules for cube roots:** The cube root function $$\sqrt[3]{x}$$ is defined

1. **State the problem:** Given that $P$ varies directly as $Q$ and the cube of $R$, find the values of $x$ and $y$ from the table where $P=0.012$, $Q=0.5$, $R=0.2$ for the first s

1. **State the problem:** Solve the equation $$4x^2 + 12x + 9 = 0$$ graphically by drawing the graph of $$y = 4x^2 + 12x + 9$$ for $$-4 \leq x \leq 2$$.

1. **State the problem:** Find the value of $x$ such that $f(x) = \sqrt[3]{x - 2} = 0$. 2. **Recall the formula and rule:** The cube root function $\sqrt[3]{y}$ equals zero if and

1. El problema es graficar la función $$f(x) = \sqrt[3]{x} - 5$$ y marcar cinco puntos: uno donde $$\sqrt[3]{x} = 0$$ y dos puntos a la izquierda y dos a la derecha de ese punto. 2

1. **State the problem:** We are given two functions: (i) $y = e^{x+1} - 1$

1. **Énoncé du problème :** Pour un projet scolaire, Émile retourne des bouteilles consignées. Il reçoit 0,10 par bouteille. Il veut gagner 25. Combien de bouteilles doit-il retour

1. **State the problem:** Simplify or expand the expression $ (2x-1)^3 $. 2. **Formula used:** The cube of a binomial $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$.