🧮 algebra

1. **State the problem:** Marsha has 300 money units to spend on paper, pencils, and pens. Prices are:

1. We are given that $p$ and $q$ are prime numbers with $p > q$, and their least common multiple (LCM) is 319. 2. Since $p$ and $q$ are prime, their LCM is simply their product: $$

1. نبدأ بشرح القاسم المشترك الأكبر (GCD) لعددين: هو أكبر رقم يقسم العددين بدون باقي. 2. ثم نوضح معنى أن العددين أوليان فيما بينهما: يعني أن القاسم المشترك الأكبر لهما هو 1.

1. **Stating the problem:** We are given two functions: - $f(x) = x^2 - 6x + 5$

1. The problem is to find the $n^{th}$ term rule for the quadratic sequence: -4, -1, 4, 11, 20, ... 2. Calculate the first differences:

1. لنفترض أن الدالة الزوجية هي $f(x)$. 2. بما أن الدالة زوجية، فإنها تحقق العلاقة $f(-x) = f(x)$ لجميع قيم $x$.

1. **تمرين الأول - تعريف وحساب الدالة h = g \circ f** نُعطى دالتين f و g مع جداول قيمهما.

1. We are asked to analyze and understand the function $$h(x) = \ln(x) + 6$$. 2. The function consists of the natural logarithm $$\ln(x)$$ which is defined for $$x > 0$$, shifted v

1. **مشكلة التمرين:** لدينا دالتين عددتين $f$ و $g$ معرفتين على جداول القيم: - \(x: -4, 0, 3\) و \(g(x): 1, 3, 0\)

1. **State the problem:** Find the domain and range of the function $$f(x) = \frac{2}{3}x + 1$$. 2. **Determine the domain:** This function is a linear function and is defined for

1. Soit $A = (2x - 3)(-x + 4) - (4x - 6)(2x - 1) + (3 - 2x)(3x + 8)$. \nDéveloppons chaque produit:\n$(2x - 3)(-x + 4) = 2x \times (-x) + 2x \times 4 - 3 \times (-x) - 3 \times 4 =

1. We are given that 40 subtracted from 60% of a number results in 50. 2. Let the number be $x$.

1. **State the problem:** Solve the equation $$\frac{2(x + 1)(2x - 2)}{3} = 3$$. 2. **Simplify the expression:** First, expand the terms inside the parentheses.

1. **Stating the problem:** We have a cubic equation $$x^3 + 2x^2 + 3x + 1 = 0$$ with roots $$\alpha, \beta, \gamma$$. We want to find:

1. مسئله: حل معادله درجه دوم $$x^2 - 5x + 6 = 0$$. 2. مراحل حل:

1. **Problem statement:** Given a quadratic equation with roots $\alpha$ and $\beta$ satisfying $$x^2 + 3x + 7 = 0,$$ find the quadratic equation whose roots are $\alpha - 2\beta$

1. Stated problem: Find the remainder when the polynomial $$2x^3 + 3x^2 - 2x + 2$$ is divided by $$x+3$$. 2. According to the Remainder Theorem, the remainder of a polynomial $$f(x

1. Stating the problem: Decompose the rational function $$F(X) = \frac{1}{X^{3}(X^{2} - 1)(X^{2} + 1)}$$ into partial fractions over $\mathbb{R}(X)$. 2. Factor the denominator: Not

1. Énoncé du problème 1.1 : Résoudre dans ℝ les systèmes (E1) : $ (x+1)^2 + y + x -1=0 $ et (E2) : $ (x+y)^2 = 1 $, puis (E1) alternatif : $ \frac{x}{1} + x + y -1=0 $ et (E2) : $

1. Problemet handlar om att beräkna hur många procent räntesatsen höjdes när den tidigare räntan var 0,5% och den höjdes med 0,2 procentenheter. 2. Vi börjar med att skriva den gam

1. **State the problem:** We are given two inequalities involving variables $\beta_{ji}$, $\theta_{ji}$, $x_{ij}$, and constants $b_{ij}$, $A_{ij}$: $$\beta_{ji} x_{ij} \leq \theta