🧮 algebra

1. **Problem:** Factorize completely the expression $x^4 - 81$. 2. **Formula and rules:** Recognize that $x^4 - 81$ is a difference of squares because $x^4 = (x^2)^2$ and $81 = 9^2

1. **Problem Statement:** Factorize the expression $b^2 - 4$. 2. **Formula Used:** This is a difference of squares, which follows the rule:

1. **State the problem:** Factorize the expression $x^2 - 1$. 2. **Recall the formula:** The expression $x^2 - 1$ is a difference of squares, which follows the rule:

1. **State the problem:** Solve the equation $x = 3x - 7$ for $x$. 2. **Write down the equation:**

1. **State the problem:** Solve the linear equation $4x + 6 = 6x$ for $x$. 2. **Recall the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation

1. **State the problem:** Factorize completely the expression $$x^2 (b - c) + 2x (b - c) + (b - c)$$. 2. **Identify common factors:** Notice that each term contains the factor $b -

1. **State the problem:** We have a function $f$ defined for all real numbers such that $$f(x + y) = f\left(\frac{x}{2}\right) + f\left(\frac{y}{2}\right)$$ for all real $x, y$. We

1. **State the problem:** Sketch the graph of the polynomial function $$y = 3x^4 + 4x^3 + 1$$ and analyze its critical points. 2. **Find the value at $$x=0$$:**

1. Problema ne cere să găsim cifra de pe poziția 40 în șirul format astfel: 1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999. 2. Observăm că șirul este format din grup

1. The problem is to simplify the expression $$\frac{3}{4} \times \frac{3}{3} \cdot \frac{2}{1}^0 \times \frac{1}{3} \times \frac{1}{3} \times \frac{2}{1}$$. 2. Recall the rules:

1. Problema este să rezolvăm o ecuație sau expresie algebrică fără a folosi fracții. 2. Pentru a elimina fracțiile, înmulțim ambele părți ale ecuației cu cel mai mic multiplu comun

1. Problema afirmă că produsul a trei numere naturale este 128. 2. Fiecare factor se înjumătățește, adică fiecare număr se împarte la 2.

1. The problem is to represent the expression $5(x+3)$ pictorially. 2. This expression means 5 times the quantity $(x+3)$.

1. **State the problem:** Simplify the expression $6 + 8 - 4 \times 6 + 8$. 2. **Recall the order of operations:** According to PEMDAS/BODMAS, multiplication is performed before ad

1. The problem asks to find the formula for the function $f$ given a hint that it can be expressed in the form $a(x - c)(x - d)$, which is a factored quadratic form. 2. The graph d

1. **State the problem:** Simplify the equation $$\frac{65 n!}{(n - 5)! 5!} = \frac{2 (n - 1) (n + 1)!}{(n - 5)! 6!}$$ and solve for $n$. 2. **Rewrite the equation:** Since $(n - 5

1. **Problem statement:** We have two functions:

1. **State the problem:** Solve the equation $$(x+1)(x+3)(x+5)(x+7) = 9$$ for $x$. 2. **Use a substitution to simplify:** Notice the terms are symmetric around the midpoint between

1. Enunțul problemei: Avem un număr natural de trei cifre \(\overline{abc}\) cu proprietățile \(c - a = 8\) și \(c - b = 3\). Trebuie să găsim valoarea numărului natural \(k\) astf

1. **Stating the problem:** We are given two sequences and asked to find the missing term in each. 2. **First sequence:** 47, 119, 167, __, 359, 527

1. مسئله را بیان می‌کنیم: اگر \(\frac{a}{5} = \frac{b}{7} = \frac{c}{11} = k\) باشد، مقدار \(\frac{abc}{(a+b+c)^3}\) را بیابید. 2. از فرض مسئله داریم: