🧮 algebra

1. **State the problem:** We have an item with an original price of $i$ dollars, and it is on sale for 25% off. We want to find expressions that correctly represent the sale price.

1. The problem is to solve for $x$ in the equation given in the document Midterm Exam-S 2.docx. Since the exact equation is not provided, please provide the specific math problem o

1. Найдите значение выражения: а) $28 \cdot 32^{5/2}$. 2. Формула для степени с рациональным показателем: $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$ где $m$ и $n$ — целые числа,

1. State the problem: Find the limit $$\lim_{x\to 3}\frac{x^2-9}{x-3}.$$ 2. Factor the numerator using difference of squares: $$x^2-9=x^2-3^2=(x-3)(x+3).$$

1. **Problem:** Evaluate the limit $\lim_{x\to 3}\frac{x^2-9}{x-3}$. 2. **Formula / idea:** When direct substitution gives $\frac{0}{0}$, factor the numerator and cancel the common

## Problem: Factor $x^2+5x+6$. 1. **Use the factoring (AC) method for $x^2+bx+c$**.

1. State the problem: Solve the linear system for $x$ and $y$. 2. Write the system of equations:

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** Solve the system of linear equations: $$2x + y = 8$$

1. **State the problem:** We are given the system of equations: $$2x + 3y = 7$$

1. The problem is to find the formula for the sum of the squares of the first $n$ natural numbers, i.e., calculate $\sum_{i=1}^n i^2$. 2. The formula for the sum of squares is:

1. **Stating the problem:** We want to understand what happens if you keep squaring the number 7 repeatedly. 2. **Formula used:** Squaring a number means multiplying it by itself.

1. The problem appears to involve simplifying and evaluating expressions involving multiplication and division of integers and fractions. 2. Let's clarify the operations step-by-st

1. The problem asks us to divide fractions and express the answers in lowest terms. 2. Recall the rule for dividing fractions: \n\n$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \tim

1. **State the problem:** Find the equations of the two lines that pass through the point $(1, 2)$ and are at a distance of $\sqrt{13}$ from the point $(6, 1)$. 2. **Set up the lin

1. **State the problem:** We need to find the total cost for 5 visits to the history museum for a member, given a linear relationship between the number of visits and total cost. 2

1. **State the problem:** We need to find the value of $x$ given by the formula: $$x=\frac{(3\times14.5)+(5\times24.5)+(8\times34.5)+(10\times44.5)+(12\times54.5)+(7\times64.5)+(3\

1. The problem asks to subtract fractions: first \(\frac{1}{8} - \frac{3}{9}\), then \(\frac{5}{6} - \frac{2}{7}\). 2. To subtract fractions, use the formula:

1. Planteamos el problema: Una bacteria se reproduce según la expresión $2^t$, donde $t$ es el tiempo en horas. 2. Queremos encontrar el tiempo $t$ cuando hay 1,024 bacterias, es d

1. **State the problem:** Find the zeros of the function $$f(x) = \frac{x^2 + 14x + 49}{x^2 + 1}$$. 2. **Recall the rule for zeros of a rational function:** The zeros occur where t

1. **Problem Statement:** Find the composite function $ (f \circ g)(x) $ where $ f(x) = \frac{x+1}{x-3} $ and $ g(x) = x + 3 $. Also, determine the domain of $ (f \circ g)(x) $.