🧮 algebra

1. ننص مشكلة السؤال: نريد إيجاد قيم $k$ بحيث يكون للمعادلة $$-3x^2 + (2k + 1)x - 4k = 0$$ جذور حقيقية. 2. لكي تكون المعادلة التربيعية لها جذور حقيقية، يجب أن يكون المميز $\Delta \g

1. Problem: Evaluate the expression $$\frac{22^2 + 22 - 12}{22 + 8}$$ and find the composite function $f \circ g(3)$ given \(g(x) = x^2\) and \(f(x) = 2x + 1\). 2. Simplify the num

1. We are asked to prove that for all $n \in \mathbb{N}$, the sum $\sum_{k=0}^{n-1} (2k+1)$ equals $n^2$. 2. Rewrite the sum explicitly to understand it better:

1. **Problem 1**: Determine if there exist non-negative quantities $x_1, x_2, x_3$ of Products 1, 2, 3 such that the labor hours in departments A, B, C fully use the monthly capaci

1. Problem 17 (a): Simplify $\sqrt{98}$. We factor 98 as $98 = 49 \times 2$.

1. State the problem: Solve the equation $4x - 28 = 0$ using the quadratic formula. 2. Rearrange the equation as a quadratic form $ax^2 + bx + c = 0$. Since there is no $x^2$ term,

### Exercise 03: Prove using mathematical induction #### 1. Prove that for all $n \in \mathbb{N}$, $\sum_{k=0}^{n-1} (2k+1) = n^2$.

1. The problem: Solve the quadratic equation $2x^2 - x - 1 = 0$ using the quadratic formula. 2. The quadratic formula is:

1. Simplify \(\sqrt{98}\): \(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}\).

1. **Énoncé du problème** : On considère la suite $(s_n)$ définie par $$s_n = 1 + \frac{1}{3} + \frac{1}{3^2} + \cdots + \frac{1}{3^n}.$$

1. Se plantea resolver cada ecuación logarítmica dada paso a paso. **a)** $$\log_2 2 + \log(11 - x^3) = \frac{2}{\log(5 - x)}$$

1. The problem states the quadratic function as $A(x) = ax^2 + bx + c$. 2. This is a general form of a quadratic equation where $a$, $b$, and $c$ are constants and $x$ is the varia

1. We are given the quadratic function $F(x) = x^2 - 6x + 5$. 2. We want to express it in the form $F(x) = (x + a)^2 + b$ where $a$ and $b$ are constants.

1. The problem is to factor the quadratic polynomial $2x^2 - x - 1$. 2. We look for two numbers that multiply to $2 \times (-1) = -2$ and add to $-1$ (the coefficient of $x$).

1. **مشكلة:** تقاسم ثلاثة إخوة أرباح تجاراتهم \n- الأول أخذ \( \frac{1}{5} \) من الأرباح \n- الثاني أخذ \( \frac{3}{10} \) من الأرباح \n- الثالث أخذ \( \frac{1}{4} \) مما أخذ الأول

1. **Definición del logaritmo:** El logaritmo de un número $x$ en base $b$ se define como el exponente al que hay que elevar $b$ para obtener $x$. Esto se expresa como: $$\log_b x

1. State the problem: Express $$\frac{(\log m)^p \sqrt{3 - n}}{m n^2}$$

1. **State the problem:** Express $$\frac{\log m^p \sqrt{3-n}}{mn^2}$$ in terms of $$\log m$$, $$\log n$$, and $$\log p$$. 2. **Rewrite the numerator:**

1. Stating the problem: We are given the function $F(x) = x^2 - 6x + 5$ and need to analyze it. 2. Identify the type of function: This is a quadratic function in standard form $F(x

1. **State the problem:** Find the domain and range of the function $$y=\frac{\sqrt{x^2 - 5x + 6}}{\sqrt{x^2 - x + 6}}.$$\n\n2. **Domain analysis:** The domain consists of all $x$

1. The problem is to analyze the quadratic function $F(x) = x^2 - 6x + 5$. 2. First, find the vertex to understand the function's shape. The vertex formula for $x$ is given by $x =