🧮 algebra

1. **State the problem:** Solve the equation $$\frac{1-\sqrt{2}}{2-\sqrt{2}} = \frac{\sqrt{3}x}{3\sqrt{3}}$$ and express the answer in surd form. 2. **Simplify the right side:** Si

1. The problem is to find the value of $A$ given the initial condition $y(1) = 2$. 2. We need the equation or function involving $A$ and $y(x)$ to proceed. Since it is not provided

1. **Stating the problem:** Find the largest integer $n$ such that

1. Masalah: Tentukan bilangan bulat terbesar $n$ sehingga $$\frac{3+9+18+30+\cdots+n}{n} < 2021.$$\n\n2. Perhatikan deret di pembilang: 3, 9, 18, 30, ...\nDeret ini bukan aritmatik

1. **State the problem:** Simplify the expression $$\frac{x^2 + 2x + 1}{x + 3}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor the numerator and d

1. **State the problem:** Given points P(3, 0), Q(-1, 6), R(k, -6), and S(4, -7), with PQ parallel to RS, find the value of $k$. 2. **Recall the formula for slope:** The slope $m$

1. **State the problem:** Divide the expression $\frac{x^2 + x}{x}$ and set it equal to 10. 2. **Use the division rule:** When dividing polynomials, divide each term in the numerat

1. **State the problem:** Solve the equation $x^3 - x^2 = 80$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. مسئله: حل معادله $$f(x) = x^3 - x - 1 = 0$$ 2. معادله را به صورت $$x^3 - x - 1 = 0$$ داریم.

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.