🧮 algebra

1. **Énoncé du problème :** Déterminer le domaine de définition $D_f$ de la fonction $f(x) = \frac{x^2 + 3x + 6}{x + 1}$.

1. **Stating the problem:** Calculate the value of $$A = \frac{3 - \frac{4}{3}}{3 + \frac{4}{3}}$$ and express it in simplified form. 2. **Formula and rules:** To simplify a fracti

1. **State the problem:** Calculate the value of $\sqrt{50} \times 3$. 2. **Recall the formula and rules:** The square root of a number $a$ is a value that, when multiplied by itse

1. The problem is to solve the equation or system given by "solve all" which is ambiguous, so we interpret it as solving a general algebraic equation. 2. To solve an algebraic equa

1. مسئله: حل معادله $1 + x = 2 \sqrt{2x - 1}$ و تعیین قابل قبول بودن ریشه‌ها. 2. فرمول و نکات مهم:

1. We are asked to simplify the expression $$\sqrt[8]{(a-4)^4}$$. 2. Recall the rule for radicals and exponents: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$.

1. Masalah: Hitung jumlah semua faktor dari bilangan $2^4 \times 5^2$. 2. Rumus: Jika sebuah bilangan memiliki faktorisasi prima $n = p^a \times q^b$, jumlah semua faktornya adalah

1. The problem is to analyze the given sets of polynomial expressions and understand their relationships or properties. 2. Let's examine each set:

1. Masalah: Hitung jumlah semua faktor genap dari $2^4 \times 5^2$. 2. Faktor dari $2^4 \times 5^2$ berbentuk $2^a \times 5^b$ dengan $a = 0,1,2,3,4$ dan $b = 0,1,2$.

1. **State the problem:** We have the system of equations: $$ax - y = 0$$

1. **Problem:** Discuss why any group is isomorphic to a group of permutations. **Explanation:** This is Cayley's theorem.

1. **State the problem:** Find the equation of the line passing through the points $(-4,7)$ and $(-4,-3)$. 2. **Formula used:** The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1

1. **State the problem:** We have 100 students with an average weight of 59 kg. The average weight of boys is 65 kg and that of girls is 50 kg. We need to find the number of boys a

1. **State the problem:** Simplify or factor the quadratic expression $2x^2 + 3x - 4$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers

1. **Énoncé du problème :** Soit $E$ un espace vectoriel réel de dimension finie $n \geq 2$ et $f$ un endomorphisme de $E$ tel que $f^2 = f$ (c'est un projecteur).

1. **State the problem:** We are given the system of equations: $$2x - ky = 14$$

1. **State the problem:** Factorize the expression completely: $$(x+3)^2 - 5(x^2 - 9)$$ 2. **Expand the terms:**

1. **State the problem:** Solve for the interest rate $r$ in the equation $$1000(1+r)^{10} = 2000.$$ 2. **Rewrite the equation:** Divide both sides by 1000 to isolate the exponenti

1. **State the problem:** Simplify the expression $$\frac{x^2 - 9}{x^2 - 6x + 9}$$. 2. **Recall the formulas and rules:**

1. **State the problem:** Find the equation of the line passing through the point $(3,4)$ with slope $-0.50$. 2. **Formula used:** The point-slope form of a line is given by:

1. The problem is to understand and simplify the expression $$\sqrt[4]{(5-a)^2}$$ given that $$a > 5$$. 2. The expression is a fourth root of a square: $$\sqrt[4]{(5-a)^2}$$.