🧮 algebra

1. **Problem statement:** Given a parabola with vertex at the origin and equation $$Ay^2 = (3A - 4)x$$ where $$A \in \mathbb{R}$$, find the value of $$A$$ such that the parabola pa

1. **Problem Statement:** Given that $\log a = 0.30$ and $\log b = 0.45$, find the value of $\log \frac{b^2}{a}$. 2. **Formula and Rules:** We use the logarithm properties:

1. **State the problem:** Solve the equation $2x + 1 = x + 5$ for $x$. 2. **Write down the equation:**

1. **State the problem:** Find the equation of the line that passes through the points $(0,-5)$ and $(4,3)$. 2. **Formula used:** The slope $m$ of a line passing through points $(x

1. نبدأ بحل المعادلة الأولى: $ (x + 1)(x + 4) = 0 $. 2. نطبق خاصية حاصل ضرب يساوي صفر: إذا كان $a \times b = 0$، فإن $a=0$ أو $b=0$.

1. **State the problem:** Factorize the quadratic expression $6x^2 - 17x - 3$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that mu

1. **State the problem:** Simplify the expression $4x - x + 2 - 3$. 2. **Combine like terms:** The terms with $x$ are $4x$ and $-x$. The constant terms are $2$ and $-3$.

1. **State the problem:** Solve the equation $10x - 1 = 4$ for $x$. 2. **Write down the equation:**

1. **State the problem:** Factorize the quadratic expression $3x^2 + 10x + 8$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that mu

1. The problem is to calculate the expression $$\frac{3x^2 - 4y^2}{3x^2 + 4y^2}$$ for given values of $x$ and $y$ or to simplify it if no values are given. 2. The formula is a rati

1. Prove that $a^4 + b^4 > 2a^2 b^2$ for any real numbers $a, b$. Start with the identity for squares:

1. The problem is to find the equation of a line given certain conditions (e.g., points or slope). 2. The general formula for the equation of a line is $y = mx + b$, where $m$ is t

1. The problem is to find the sum of the infinite geometric series given by the sequence. 2. The formula for the sum $S$ of an infinite geometric series with first term $a$ and com

1. **Problem Statement:** Complete the 3x3 magic square such that the sum of the numbers in each row, column, and diagonal equals 21. Given partial magic square:

1. **Problem statement:** A green light flashes every 12 minutes, and a red light flashes every 45 minutes. Both lights flash together at 9:00 AM. We need to find the next time whe

1. **State the problem:** Solve the linear equation $-17x + 10 = -7x + 140$ for $x$. 2. **Write down the equation:**

1. **Stating the problem:** A green light flashes every 12 minutes. We want to understand the timing pattern of these flashes. 2. **Formula and explanation:** The green light flash

1. **State the problem:** Solve the equation $-7x + 10 = -7x + 140$ for $x$. 2. **Write down the equation:**

1. Simplify $\frac{\sqrt[3]{m^{3}n^{6}} \times m^{4}n^{2}}{(\sqrt{m^{2}})^{2}n^{3}}$. Rewrite the cube root: $\sqrt[3]{m^{3}n^{6}} = m^{3/3}n^{6/3} = m^{1}n^{2}$.

1. **State the problem:** Factor the given expression. Since no specific expression was provided, let's consider a general approach to factoring. 2. **Formula and rules:** Factorin

1. **Stating the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y + z = 4 \\ 2x - y + 3z = 1 \\ -x + 3y + 2z = 7 \end{cases}$$