🧮 algebra

1. **Énoncé du problème** : Résoudre dans $\mathbb{R}^+$ les équations du second degré suivantes : 1) $x^2 - 5x + 6 = 0$

1. The problem involves analyzing and understanding the given functions and their properties. 2. Let's examine each function one by one.

1. مسئله: حل معادله $2x + 3 = 7$ را داریم. 2. فرمول و قوانین: برای حل معادله‌های خطی، هدف این است که $x$ را تنها کنیم. برای این کار، ابتدا باید اعداد ثابت را از طرف $x$ جدا کنیم.

1. The problem involves evaluating the sum $$\sum_{k=1}^{n+1} a + k C_{k-1} d$$. 2. First, clarify the expression: it appears to be a sum from $k=1$ to $n+1$ of the terms $a + k \c

1. **Problem Statement:** Solve the summation $$\sum_{k=1}^{n+1} a_k C_{k-1} d$$ where $a_k$, $C_{k-1}$, and $d$ are terms given or to be defined.

1. **State the problem:** We need to find the sum of the series given by $$\sum_{k=1}^{n+1} (a + C (k-1) d)$$ where $a$, $C$, and $d$ are constants.

1. **State the problem:** Find the equations of the line of intersection of the planes given by $$x + 2y + 3z = 6$$

1. **State the problem:** Simplify the expression $5y + (y-2)(y+8)$.\n\n2. **Recall the formula:** Use the distributive property (FOIL) to expand $(y-2)(y+8)$.\n\n3. **Expand the p

1. **State the problem:** Calculate the sum $$\sum_{x=1}^6 (-1)^x 3^x$$. 2. **Formula and explanation:** This is a finite sum of terms where each term is $$(-1)^x 3^x$$. We can wri

1. The problem is to evaluate the sum $$\sum_{x=1}^5 (-1)^x 3^x$$. 2. The formula for the sum of terms is given by adding each term from $x=1$ to $x=5$:

1. **Problem:** Prove by mathematical induction that $$4 + 7 + 10 + \cdots + (3n + 1) = \frac{n(3n+5)}{2}$$ for all positive integers $n$. 2. **Base Case:** For $n=1$, the left sid

1. **State the problem:** We need to evaluate the sum $$\sum_{x=2}^5 (x - 2x + 1)$$. 2. **Simplify the expression inside the sum:**

1. **State the problem:** Solve the system of equations using the elimination method: $$7x - 5y = 26$$

1. **State the problem:** We need to graph the line given by the equation $$y = -\frac{2}{3}x + 1$$ on a coordinate grid. 2. **Understand the equation:** This is a linear equation

1. Kita diminta menentukan titik pusat dan jari-jari lingkaran dari persamaan yang diberikan. 2. Rumus umum lingkaran adalah $$(x - h)^2 + (y - k)^2 = r^2$$ dimana $(h,k)$ adalah t

1. **State the problem:** Simplify the expression $$\frac{2np^5r^7}{16p^{20}r}$$ without negative exponents. 2. **Write the formula and rules:** When dividing powers with the same

1. **State the problem:** Solve for $x$ in the equation $$3^x + 2^x = 13.$$\n\n2. **Understand the equation:** We want to find the value of $x$ such that the sum of $3^x$ and $2^x$

1. **State the problem:** Simplify the expression $$\frac{(5n^8p^6)^2}{30n^4p^2}$$. 2. **Apply the power to each factor inside the parentheses:**

1. The problem is to prove a mathematical statement using equations. 2. Let's consider a simple example: proving the distributive property of multiplication over addition, which st

1. **State the problem:** Solve the linear equation $7x - 5y = 26$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation. Thi

1. **State the problem:** Solve for $x$ in the equation $$3^x + 2^x = 13.$$\n\n2. **Understand the equation:** We have an exponential equation with two terms, $3^x$ and $2^x$, summ