🧮 algebra

1. **State the problem:** Solve the cubic equation $$f(x) = x^{3} - x - 1 = 0$$ for the roots. 2. **Formula and rules:** Cubic equations can have one or three real roots. We can us

1. The problem asks to find the domain and range of the function given by the set of points $\{(1,5), (3,-2), (5,1), (7,-5)\}$. 2. The domain of a function is the set of all possib

1. The problem is to understand the formula for the sum of a sequence: $S_n = U_1 + U_2 + \cdots + U_{n-1}$. 2. Here, $S_n$ represents the sum of the first $n-1$ terms of a sequenc

1. **Problem statement:** Michael sells 35 cabbage rolls with a profit of 60%. He sells 5 rolls for 20. We need to find the total cost of raw materials for all 35 rolls. 2. **Formu

1. **Stating the problem:** Michael sells 35 cabbage rolls with a 60% profit. He sells 5 rolls for 20. We need to find the total cost of raw materials for all 35 rolls. 2. **Unders

1. The problem is to evaluate $2^3$. 2. The expression $2^3$ means 2 raised to the power of 3, which is 2 multiplied by itself 3 times.

1. Let's start with a problem on quadratic equations: Solve for $x$ in the equation $$x^2 - 5x + 6 = 0.$$ 2. The formula to solve quadratic equations is the quadratic formula: $$x

1. مسئله: تعیین دامنه و برد تابع $$f(x) = 7\sqrt{4 - x^2}$$. 2. دامنه تابع: چون داخل رادیکال باید بزرگ‌تر یا مساوی صفر باشد، شرط زیر را داریم:

1. مسئله: حل معادله $2x^2 - 5x + 3 = 0$ است. 2. فرمول استفاده شده: معادله درجه دوم به صورت کلی $ax^2 + bx + c = 0$ است و ریشه‌ها با فرمول

1. مسئله: تعیین دامنه و برد تابع $f(x) = \sqrt{x^2 - 9}$ است. 2. دامنه تابع: برای اینکه مقدار زیر رادیکال منفی نشود، باید داشته باشیم:

1. You asked for more questions on further math. Let's consider a common algebra problem: Solve the quadratic equation $ax^2 + bx + c = 0$. 2. The formula to find the roots of a qu

1. مسئله: تعیین دامنه و برد تابع $f(x) = \sqrt{x + 3}$ است. 2. دامنه تابع‌های رادیکالی با ریشه زوج (مثل ریشه دوم) شامل مقادیری از $x$ است که زیر رادیکال منفی نباشد.

1. **State the problem:** Simplify the expression $5(x+y)-2(x-y)$. 2. **Use the distributive property:** Multiply each term inside the parentheses by the factor outside.

1. **State the problem:** Valeria spent a total of 21 on 8 games and rides. Each game costs 1.50 and each ride costs 3. We want to find the number of games $x$ and rides $y$. 2. **

1. **State the problem:** A jeweler has a fixed amount of gold and makes bracelets and necklaces. Each bracelet uses 7 grams of gold, each necklace uses 18 grams, and the total gol

1. **State the problem:** Solve the inequality $$5h + 5(9h + 1) \leq 5h + 1 - 2$$ for $h$. 2. **Apply the distributive property:** Multiply $5$ by each term inside the parentheses:

1. **Problem:** Find the first 4 terms in the power series expansion of $$f(x) = \frac{(1+2x)^2}{(1-x)^2}$$ and state when the expansion is valid. 2. **Formula and rules:**

1. The problem is to find the square root of 81. 2. The square root of a number $x$ is a value $y$ such that $y^2 = x$.

1. The problem is to simplify the expression $$\frac{n \times n + n}{n}$$ and find its value when $n=10$. 2. The formula involves algebraic simplification of fractions. Important r

1. **State the problem:** Solve the equation $x^2 = 16$ for $x$. 2. **Formula and rules:** To solve equations of the form $x^2 = a$, where $a$ is a positive number, we use the squa

1. **State the problem:** Solve the equation $x^2 + 27 = -12x$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: