🧮 algebra

1. **Stating the problem:** We want to find the term independent of $x$ (the constant term) in the expansion of $$\left(2 + \frac{3}{x^2}\right)^{10} \left(1 - 4x^2\right)^2.$$\n\n

1. The given expression is $2x+3y$. 2. This is a linear algebraic expression involving variables $x$ and $y$ with coefficients 2 and 3 respectively.

1. **State the problem:** We want to find the function $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $m, n$, the equation $$ f(m + n f(m)) + m = f(m f(n)) + f(n f(

1. **State the problem:** Find the coordinates of the points A and B where the line $y=4x-3$ meets the curve $y=3+5x-2x^2$. Then find the area of triangle $POQ$ formed by the perpe

1. Problem 1: Solve $5x^2 = 30x$ - Divide both sides by $5$: $x^2 = 6x$

1. The problem states: Solve for $y$ in the function $y = a^x$ where $a > 0$ and $a \neq 1$. 2. This means the function represents an exponential function with base $a$.

1. The problem asks to "Give answers to blanks," but no specific blanks or expressions were provided. 2. Please provide the equations or expressions with blanks for me to solve or

1. Problem 11: Solve for $u-v$ from the system: $$3u-2v=9$$

1. Prove the commutative law of addition for matrices A and B where: $$A=\begin{bmatrix}3 & 4 \\ 2 & 5\end{bmatrix},\quad B=\begin{bmatrix}3 & 2 \\ 1 & 1\end{bmatrix}$$

1. The problem is to find the values of $X$ and $y$ from the equation $$X^2 + y^2 = (1 + xi)(3 + i)$$ where $i$ is the imaginary unit. 2. First, expand the right-hand side by multi

1. Let's first understand the context: you want a more detailed explanation of solving a math problem. 2. When solving algebraic problems, we proceed step-by-step, breaking down ea

1. We are given the first equation to solve: $$(z^2 - 2yz - y^2)p + (xy + zx)q = xy - zx.$$ 2. We are also given the second equation: $$(3x + y - z)p + (x + y - z)q = 2(z - y).$$

1. **Problem statement:** Find $2A - B$ where $A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 13 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & -3 \\ -5 & 1 \\ 4 & 2 \end{bmatrix}$.

1. We are asked to simplify the expression $\frac{999 \times 998}{999 \times 999} + 999$. 2. First, notice the fraction: $\frac{999 \times 998}{999 \times 999}$. We can cancel out

1. **Evaluate expressions using a 12-hour clock where ⊕ means addition modulo 12 and ⊖ means subtraction modulo 12.** - a. $3 \oplus 5 = (3 + 5) \mod 12 = 8$

1. The problem asks: Which parabola has its branches closest to the Oy-axis among the following options? 2. All given parabolas have the form $y=ax^2$. The distance of the parabola

1. Diberikan operasi biner $a@b = \frac{a}{b} + \frac{b}{a} - 2$. Kita akan menyelesaikan persamaan $$((c + 1)@c)@(c - 1) = (c + 1)@(c@(c - 1))$$

1. The problem states several equalities for a function $P(x)$: $P(0)=0$, $P(3)=a=P(5)$ and $P(11)=2a=P(13)$. 2. This means the function $P(x)$ takes the same value $a$ at $x=3$ an

1. Problem: Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. Step 1: Identify coefficients: $$a=1$$, $$b=-5$$, $$c=6$$.

1. Let's start by understanding the concept of a line equation. A line in a plane can be represented by the equation:

1. The equation of a line is given by $y = m \times x + c$, where $m$ is the slope and $c$ is the intercept. 2. The slope $m$ represents how steep the line is and is calculated as