🧮 algebra

1. **Problem statement:** Alan cycled from P to Q (64 km) starting at 10:00, stopped for a break, then continued to arrive at 15:00. Brendan started at 12:00 from P to Q, passing A

1. **Problem statement:** We are given a cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ with the following conditions:

1. **State the problem:** We are given the equation $$2kv(v+w) - (1 + k(v+w)^2) = 0$$ and want to understand or simplify it after differentiation. 2. **Rewrite the equation:** The

1. **Problem statement:** Vervollständigen Sie die Tabelle für ganzrationale Funktionen 3. und 4. Grades.

1. **State the problem:** We need to substitute $$t = \frac{-x + \sqrt{x^2 + 2y^2}}{2y}$$ into the equation $$x + 2ty = 8t^2 + 4$$ and show that the result simplifies to $$\frac{x}

1. **Stating the problem:** A car travels from point A to point B at an unknown speed $v$ km/h (unloaded), and returns from B to A at 60 km/h (loaded). The average speed for the en

1. **State the problem:** We are given a piecewise function defined as $$y=\begin{cases} x+6 & \text{if } -6 \leq x \leq -2 \\ x^{2} & \text{if } -2 < x < 3 \\ -x+12 & \text{if } x

1. **State the problem:** We have a piecewise function defined as $$f(x) = \begin{cases} -\frac{x}{2} + 6 & \text{for } -10 \leq x \leq -4 \\ \frac{10}{3} & \text{for } -4 < x < 3

1. Problem G 1 a) Partialbruchzerlegung von $$f(s) = \frac{1}{(s-2)(s+2)}$$ 2. Die Funktion ist ein Bruch mit zwei Linearfaktoren im Nenner. Die Partialbruchzerlegung hat die Form:

1. **Stating the problem:** We have a drug dosage problem where a 20 mg tablet is taken once daily, and the drug amount decreases by 30% each day. The amount left after $n$ days is

1. **Stating the problem:** We are given two lines and need to find the two possible values of $m$ such that the area of the triangle formed is a certain value (assumed given or to

1. **State the problem:** We are given two points $(4,0)$ and $(0,2)$ and need to find the equation of the line passing through these points. 2. **Formula used:** The slope-interce

1. **State the problem:** We are given three functions: $f(x) = 2x + 8$, $g(x) = x^2 + 2x - 8$, and $h(x) = 3x - 6$. We need to find the value of $f(0.5)$. 2. **Recall the formula:

1. The problem asks to find the domain of the function $$f(x) = \sqrt{-4x - 7}$$. 2. The domain of a square root function requires the expression inside the root to be non-negative

1. **State the problem:** We want to rewrite the expression $$(8.9 \times 10^8) - (4.4 \times 10^7)$$ so that both terms have the same power of 10, specifically $10^8$. 2. **Recall

1. **State the problem:** Given two functions $f(x) = 4 - x^2$ and $g(x) = 2 - x$, find the product function $(fg)(x)$, which means $f(x) \cdot g(x)$. 2. **Formula and explanation:

1. **State the problem:** Find the product of the functions $f(x) = 4 - x^2$ and $g(x) = 2 - x$, i.e., find $(fg)(x) = f(x) \cdot g(x)$.

1. **State the problem:** We have a manufacturing company with cost model $$C(x) = 0.35x^3 + 0.02x^2 + 4x + 240$$ and sales price model $$S(x) = 60 - 0.02x$$, where $x$ is the numb

1. **State the problem:** We are given two functions $f(x) = x^2 + 9$ and $g(x) = x + 3$. We need to determine which operation among addition, subtraction, multiplication, and divi

1. **State the problem:** We are given two functions: $f(x) = 2x + 3$ and $g(x) = 3[f(x)]^2 - 2$. We need to find the explicit form of $g(x)$ in terms of $x$. 2. **Recall the formu

1. **State the problem:** Given two functions $p(x) = ab^x$ and $r(x) = ca^x$, find the expression for $p(x) \cdot r(x)$. 2. **Recall the formula for multiplying exponential functi