🧮 algebra

1. **Постановка задачи:** Упростить выражение $$\frac{(x-3)^2}{x^3-27} : \frac{x^2-6x+9}{x^2+3x+9}$$. 2. **Запишем деление дробей как умножение на обратную:**

1. The problem involves analyzing the commuter train distance over time given points on a graph. 2. The points provided are (10,15), (20,20), (30,15), (40,0), (50,15), (60,20), (70

1. The problem asks us to find an algebraic expression for "seven times the sum of a number, n, and four." 2. The phrase "sum of a number, n, and four" means we add n and 4: $$n +

1. **State the problem:** We need to find the difference in the amounts Susan and Jim charge for 10 hours of lawn care work. 2. **Analyze Susan's charges:** From the table, Susan's

1. **Problem Statement:** We are given two sets of data representing the relationship between the number of hours worked and the amount charged by two lawn care services: Susan's L

1. Задача: упростить выражение $8x^3 + 0.064y^3$. 2. Формула: заметим, что $8x^3 = (2x)^3$ и $0.064y^3 = (0.4y)^3$.

1. Дано выражение: $ (a+(b-c))^2 $. 2. Используем формулу квадрата суммы: $ (x+y)^2 = x^2 + 2xy + y^2 $.

1. **State the problem:** Simplify the expression $$\frac{16}{32} \times \frac{-24}{72} \times \frac{-12}{-8}$$. 2. **Recall the rules:** When multiplying fractions, multiply the n

1. **State the problem:** Simplify the expression $$\frac{-16}{32} \times \frac{-24}{72} \times \frac{-12}{-8}$$. 2. **Recall the rules:** When multiplying fractions, multiply the

1. **Problem:** Simplify $5(4x + y)$. Formula: Use distributive property $a(b + c) = ab + ac$.

1. Задача: Раскрыть скобки в выражении $$(3a - b)^2 - (3a + b)^2$$. 2. Формула для квадрата разности и суммы:

1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} 4x - 5, & x \leq 2 \\ 3x - 2x^2, & x > 2 \end{cases}$$

1. Задача: разложить выражение $x^4 - 27x$ на множители. 2. Формула и правила: для разложения на множители сначала вынесем общий множитель за скобки, затем применим формулы разложе

1. **State the problem:** We need to find the value of $f$ for the point $(5, f)$ on a line passing through $(-1, 9)$ with a gradient (slope) of $-2$. 2. **Formula for gradient:**

1. Задача: упростить выражение $49x^6 - 64$. 2. Формула: это разность квадратов, которая имеет вид $$a^2 - b^2 = (a - b)(a + b)$$.

1. The problem states that the quadratic expression $a x^{2} + b x + c$ can be rewritten as $$10 (x - 3)^{2} - 23$$ by completing the square. 2. Recall the formula for expanding a

1. Задача: упростить выражение $49x^2 - 64$. 2. Формула: это разность квадратов, которая упрощается по формуле $$a^2 - b^2 = (a - b)(a + b)$$.

1. **State the problem:** We have a multiplication grid with unknowns $a$, $b$, $c$, and $d$. The grid shows products of numbers in the top row and left column. 2. **Understand the

1. **State the problem:** Given the equation $\log x + \log 3 = y \log 6 + \log 6$, show that $x = y + 1$. 2. **Recall logarithm properties:**

1. **State the problem:** We need to calculate the percentage error when the predicted temperature is 25 and the actual temperature is 22. 2. **Formula for percentage error:**

1. **State the problem:** Find the gradient (slope) of the straight line passing through the points $(14, -6)$ and $(16, 10a)$. 2. **Formula for gradient:** The gradient $m$ of a l