🧮 algebra

1. **Problem Statement:** Find the square root of a complex number $z = a + bi$, where $a$ and $b$ are real numbers. 2. **Formula and Explanation:** The square root of a complex nu

1. **State the problem:** Solve the equation $$\frac{3}{7x - 1} + \frac{1}{x} = 0$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common

1. مسئله: دنباله‌ای داده شده است که جمله اول آن $a_1 = 8$ است و رابطه بازگشتی برای جملات بعدی به صورت $$a_{n+1} = n a_n + 1$$ است. هدف یافتن چهار جمله بعدی دنباله است. 2. فرمول و ت

1. مسئله را بیان می‌کنیم: می‌خواهیم مقدار $$a_3 - (b_2 \times c_4)$$ را با توجه به تعاریف $$a_n = \frac{n^2}{(-1)^n}$$، $$b_n = 2^{-n}$$ و $$c_n = n^2 - n$$ پیدا کنیم.

1. مسئله را بیان می‌کنیم: ما سه عبارت جبری داریم: $a^3$, $c^4$, و $b^2$. 2. این عبارات توان‌های مختلفی از متغیرهای $a$, $c$, و $b$ هستند.

1. مسئله را بیان می‌کنیم: باید مقدار $a_3 - (b_4 \times c_6)$ را پیدا کنیم. 2. فرمول‌های داده شده را یادآوری می‌کنیم:

1. **State the problem:** Simplify the expression $ (c4 \times b2) - a3 $. 2. **Interpret the variables:** Assuming $c4$, $b2$, and $a3$ represent variables or terms, the expressio

1. مسئله را بیان می‌کنیم: می‌خواهیم مقدار عبارت $$3a - (b^2 \times c^4)$$ را با توجه به دنباله‌ها $$a_n = \frac{n^2}{(-1)^n}$$، $$b_n = 2^{-n}$$ و $$c_n = n^2 - n$$ پیدا کنیم.

1. **State the problem:** We need to find Mona's present age given that Sana is 18 years older than Mona and that after 6 years, Sana will be twice as old as Mona. 2. **Define vari

1. **State the problem:** Rehan buys a book and a pen and pays 50 for both. The book costs 8 more than twice the cost of the pen. We need to find the cost of the book. 2. **Define

1. **State the problem:** Simplify the expression $$\frac{0.1 + (1 - 0.1)}{1 + 2\sqrt{7}} - \frac{1}{-1 + 2\sqrt{7}} + \frac{2}{-1 + 1.9} + \frac{3}{?}$$

1. **Stating the problem:** Solve the first system of equations simultaneously:

1. The question asks for the values of $q$, $L$, and $K$. 2. To provide accurate answers, we need the context or formulas related to $q$, $L$, and $K$.

1. **State the problem:** Solve the equation $$e^{x+1} - 18e^x - 3 = 0$$ for $x$. 2. **Rewrite the equation:** Use the property of exponents: $$e^{x+1} = e^x \cdot e^1 = e \cdot e^

1. **Problem Statement:** Find a root of the function $f(x) = x^3 - x - 3$ in the given intervals. 2. **Recall the Intermediate Value Theorem:** If $f$ is continuous on $[a,b]$ and

1. **State the problem:** Solve the quadratic expression $5x^2 + 2x + 45$ for $x$. 2. **Formula used:** To solve a quadratic equation $ax^2 + bx + c = 0$, use the quadratic formula

1. The problem is to graph the linear function $y = -\frac{1}{3}x + 5$. 2. The formula for a linear function is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

1. Let's create a question based on the lesson you studied. 2. Suppose the lesson was about solving linear equations. Here's a question: Solve for $x$ in the equation $$3x + 5 = 20

1. The problem is to construct functions that model linear relationships. 2. A linear function has the form $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept.

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **State the problem:** Simplify and factorize the expression $16x^2 - 100$. 2. **Recognize the form:** This is a difference of squares since $16x^2 = (4x)^2$ and $100 = 10^2$.