🧮 algebra

1. **Problem statement:** You took $\frac{1}{3}$ of the chocolates from a box, then your brother took $\frac{3}{5}$ of the remaining chocolates. After that, 20 chocolates were left

1. **Énoncé du problème :** Trouver le nombre de solutions dans $\mathbb{C}$ de l'équation $$x^2 + a|x| + b = 0$$ où $a,b \in \mathbb{R}$.

1. **State the problem:** We are given the function $$f(x) = \frac{x^3 + 5x^2 - 6x}{x^2 - 1}$$ and need to find: (i) The vertical asymptote(s) and/or hole(s) of the graph.

1. **State the problem:** Simplify the expression $$\frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}}$$. 2. **Formula and rule:** To simplify a fraction with radicals in the denomina

1. Tarkastellaan eksponentiaalista mallia $$y = ae^{bx}$$, jossa parametrit $$a$$ ja $$b$$ ovat satunnaismuuttujia. 2. Annetut tiedot:

1. **State the problem:** Find the interval of $x$ such that $$\left|\,|x-3|-2\right| \geq 1.$$\n\n2. **Understand the expression:** The expression involves nested absolute values.

1. **Problem (a):** Find the solution set of the system of equations: $$\begin{cases} xy = 3 \\ 3x + y = 10 \end{cases}$$

1. **State the problem:** Simplify and analyze the quadratic expression $3 + 8x - 2x^2$. 2. **Identify the type of expression:** This is a quadratic polynomial in the form $ax^2 +

1. **Problem Statement:** Rewrite the repeating decimal $1.83\overline{3}$ as a simplified fraction. 2. **Understanding the repeating decimal:** The number $1.83\overline{3}$ means

1. **State the problem:** Solve the quadratic equation $$3 - 4x - 2x^2 = 0$$ for $x$. 2. **Rewrite the equation in standard form:** The standard form of a quadratic equation is $$a

1. **State the problem:** Given the equation $\log \sqrt{x} = -1$, find the value of $x$. 2. **Recall the logarithm and root properties:**

1. The problem asks whether the function $y = 2 \sin(x)$ is one-to-one. 2. A function is one-to-one (injective) if each output corresponds to exactly one input.

1. The problem asks for the domain of the function $$f(x) = \sqrt{x^2 - 5x + 6}$$. 2. The domain of a square root function requires the expression inside the root to be non-negativ

1. **State the problem:** Determine the symmetry properties of the function $$f(x) = \frac{x}{x^2 + 1}$$. 2. **Recall definitions:**

1. **Problem Statement:** Given:

1. The problem is to solve the equation or expression given by the user. However, since no specific problem was provided, I will explain how to approach solving algebraic equations

1. **State the problem:** Make $x$ the subject of the equation $fx + 3y = 4x$. 2. **Rewrite the equation:** The equation is $fx + 3y = 4x$.

1. **State the problem:** Make $x$ the subject of the equation $px - y = 3y$. 2. **Write down the equation:**

1. **State the problem:** Make $x$ the subject of the equation $x + y = 2y + 3x$. 2. **Write down the equation:**

1. **State the problem:** Simplify the expression $2\sqrt{63^3}$.\n\n2. **Recall the formula and rules:** The square root of a power can be simplified using the property $\sqrt{a^b

1. The problem is to find the cube root of 64, which means finding a number $x$ such that $x^3 = 64$. 2. The cube root is denoted as $\sqrt[3]{64}$ and can be expressed as $64^{\fr