🧮 algebra

1. **State the problem:** Simplify the rational expression given by $$\frac{x^2 - 2x + 8}{1}$$ (assuming the expression is over 1 since no denominator was provided). 2. **Identify

1. **State the problem:** Simplify the rational expression \( \frac{4y(y - 3)(y + 4)}{y(y^2 - y - 6)} \). 2. **Factor the denominator:** The quadratic \( y^2 - y - 6 \) factors as

1. Let's start by understanding what a function is. A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output

1. **State the problem:** Let $p$ be the kilograms of cheese produced.

1. Aufgabe c ist nicht explizit im Text angegeben, aber basierend auf den gegebenen Informationen und den mathematischen Ausdrücken konzentrieren wir uns auf die Berechnung von $0,

1. **State the problem:** Find the values of $a$ such that the distance between the points $(1, a)$ and $(3, 5)$ is $\sqrt{8}$.\n\n2. **Formula used:** The distance $d$ between two

1. **State the problem:** We need to find the values of $a$ and $b$ such that the midpoint of the points $(a, 3)$ and $(7, b)$ is $(5, 4)$. 2. **Formula for midpoint:** The midpoin

1. **Problem statement:** We need to find a rule for the volume $V$ (in litres) of water in a flask as a function of time $t$ (in minutes), given that the flask is filled at a cons

1. **State the problem:** Find the vertex, y-intercept, and x-intercepts of the quadratic function $$y = 3x^2 - 12x + 9$$ using calculation. 2. **Vertex formula:** The vertex of a

1. **State the problem:** Find the vertex, y-intercept, and x-intercepts of the parabola given by the equation $$y = x^2 - 4$$. 2. **Vertex:** The vertex form of a parabola is $$y

1. مسئله: عبارت جبری $4y - y - 4$ را با استفاده از تجزیه به دست آورید. 2. ابتدا باید عبارات مشابه را با هم جمع کنیم. در اینجا $4y$ و $-y$ عبارات مشابه هستند.

1. مسئله را بیان می‌کنیم: عبارت جبری $$-3ab - 2b^3c + ab^2$$ را با استفاده از تجزیه به عوامل ساده‌تر کنیم. 2. ابتدا عبارت را به صورت کامل می‌نویسیم:

1. **State the problem:** A machine costs 45000 and has a salvage value of 4350 after 6 years. The depreciation is computed using a constant percentage rate on the declining book v

1. **State the problem:** Solve for $c$ in the equation $7c = 4c - 13 + c + 36$. 2. **Combine like terms on the right side:**

1. The problem asks for the value of $c$, but no equation or context is provided. 2. To find $c$, we need an equation or relationship involving $c$.

1. **State the problem:** Solve the equation $32c + 5 = 24c + (c + 32)$ for $c$. 2. **Expand and simplify:** First, expand the right side:

1. **State the problem:** Solve for $s$ in the equation $(19s - 18) + (33s - 36) = 32s$. 2. **Combine like terms on the left side:**

1. Problem: Calculate $\log_3 x$ for given expressions of $x$ with $a,b,c>0$ and base 3. 2. Recall logarithm rules:

1. **Problem statement:** Simplify and expand the expression $5(u+5)(u-5)(u+2)$. 2. **Recall the difference of squares formula:**

1. **State the problem:** Expand the expression $2(x-4)(x+4)$. 2. **Recall the formula:** The product of two binomials of the form $(a-b)(a+b)$ is a difference of squares:

1. Problem 15 asks to find $\log_2 x$ for various expressions of $x$ where $a,b,c>0$. 2. Recall the logarithm properties: