🧮 algebra

1. **State the problem:** Paul wants to build a rectangular garden with a fence of 66 meters in total length. The length $L$ of the garden should be twice the width $W$: $$L=2W$$ 2

1. The problem involves understanding and working with complex numbers. 2. Recall that a complex number is generally of the form $z = a + bi$ where $a$ is the real part and $b$ is

1. **Stating the problem:** We are given the line equation $$y = 7x - 5$$ and a point $$(-5, -5)$$. We need to find:

1. Le problème demande de simplifier l'expression $$\frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-a)(b-c)} + \frac{c^3}{(c-a)(c-b)}$$. 2. Observons que chaque terme est de la forme $$\fra

1. Stating the problem: Find the slope of lines parallel and perpendicular to the line given by the equation $$8x + 6y = 4$$. 2. Rewrite the equation in slope-intercept form $$y =

1. **State the problem:** We need to find the equation of the line passing through points $(1, -1)$ and $(5, 1)$. 2. **Find the slope (m):** The slope formula is $$m = \frac{y_2 -

1. Stating the problem: We need to find the equation of a line in slope-intercept form $y = mx + b$ that passes through the point $(10, -9)$ and has slope $m = -\frac{1}{2}$. 2. Us

1. The problem is to find the value of the expression $$4 \times \frac{9}{10} + \frac{3}{11} + \frac{7}{15}$$. 2. First, calculate each part:

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.