🧮 algebra

1. Problem: Given positive integers $a, b, c, d$ satisfy the equation $$a + \frac{1}{b + \frac{1}{c + \frac{1}{d}}} = \frac{23}{16}.$$ Find the value of $a + b + c + d$. 2. First,

1. **State the problem:** Find the sum to infinity of the geometric progression (GP) 48, 12, 3, 0.75, ... 2. **Identify the first term and common ratio:** The first term $a = 48$.

1. **State the problem:** Find all zeros of the polynomial function $$f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24$$. 2. **Recall the method:** To find zeros of a polynomial, we try to fac

1. **State the problem:** "3 added to x is 5" means $x + 3 = 5$. 2. **Write the equation:** $x + 3 = 5$

1. **State the problem:** Find the sum to infinity of the geometric progression (GP) 16 + 8 + 4 + 2 + ... 2. **Formula for sum to infinity of a GP:**

1. The problem is to solve the equation $42 \div o = 7$ for $o$. 2. The formula to solve for $o$ when dividing is $o = \frac{42}{7}$.

1. **State the problem:** We are given the function $f(x) = \sqrt{x-4}$ and asked to understand its domain, range, and graph. 2. **Domain:** The expression inside the square root m

1. **Problem:** Find the domain and range of the function $f(x) = \sqrt{x - 2}$ and create a table of values. 2. **Formula and rules:** The square root function $\sqrt{u}$ is defin

1. **State the problem:** Solve the system of linear equations: $$3x + 4y = 14$$

1. **State the problem:** We are given the 2nd and 5th terms of a geometric progression (GP) as $-6$ and $48$ respectively. We need to find the sum of the first four terms. 2. **Re

1. **State the problem:** Solve the equation $x^2 - 5x + 6 = 0$. 2. **Formula and rules:** To solve a quadratic equation $ax^2 + bx + c = 0$, we can factor it, complete the square,

1. The problem asks to explain why $$e^{\ln|x|} = |x|$$ holds true. 2. Recall the definition of the natural logarithm function \(\ln y\) as the inverse of the exponential function

1. **State the problem:** Ms. A owns a house worth 10000. She sells it to Mr. B at a 10% profit, then Mr. B sells it back to Ms. A at a 10% loss. We need to find how much money Ms.

1. **State the problem:** We are given two functions: $$f(x) = \frac{e^{2x} + 2e^{x} + 2}{e^{x} + 1}$$

1. **State the problem:** Solve the simultaneous equations for variables $x$ and $y$. 2. **General approach:** Simultaneous equations are solved by either substitution or eliminati

1. **State the problem:** Solve the equation $\tan(xy) = e^{xy} + x$ for variables $x$ and $y$. 2. **Understand the equation:** The equation involves a transcendental function $\ta

1. The problem is to understand the variables $\omega$ and $z$ and their possible relationships or uses. 2. In mathematics and physics, $\omega$ often represents angular frequency

1. **State the problem:** Solve the simultaneous linear equations for variables $x$ and $y$. 2. **General form:** A system of two linear equations can be written as:

1. **Énoncé du problème :** Vérifier si 0 est une racine du polynôme $P(x) = 2x^2 + 7x + 2$.

1. **State the problem:** Solve the equation $5(115 - n) = 3n - 17$ for $n$. 2. **Apply the distributive property:** Multiply 5 by each term inside the parentheses.

1. **State the problem:** Two dogs start 500 feet apart and run toward each other at 10 ft/sec each. A flea flies back and forth between them at 25 ft/sec until the dogs collide. W