🧮 algebra

1. The question "when do we half the number" relates to situations where a quantity is reduced to half its original value. 2. This commonly appears in problems involving exponentia

1. The problem: Find the equation of an ellipse given its vertices. 2. Recall the standard form of the ellipse equation centered at the origin with horizontal major axis:

1. **Problem 20:** Given height equation $$h = -16t^2 + 96t$$ , find $$t$$ when $$h = 144$$ feet. 2. Substitute $$h = 144$$:

1. 問題陳述：已知 $x,y,z$ 均為正實數，且滿足條件 $$x^{2}+y^{2}+z^{2}=1$$ 求表達式 $$\sqrt{2}xy + yz$$ 的最大值。

1. Problem statement: Evaluate $y$ for the linear function $y = -\frac{1}{3}x - \frac{2}{5}$ at $x = 0$, and describe the graph. 2. Substitute $x = 0$ into the formula to get $y =

1. **Problem statement:** Given total monthly profit Rs. 250000, monthly revenue is twice the monthly cost minus Rs. 60000. We need to find monthly revenue and monthly cost. 2. Let

1. **State the problem:** We are given the price per product unit as Rs. 1200, delivery fee per product as Rs. 200, and a discount of 4% applied if the total amount exceeds Rs. 500

1. **প্রশ্ন:** (a) যোগফল নির্ণয় কর

បញ្ហា: ដោះស្រាយសមីការ $x^2 - 5x + 6 = 0$. 1. យើងចាប់ផ្តើមដោយពិនិត្យថាតើអាច factor សមីការនេះបាន ឬទេ។

1. **State the problem:** We need to express the variable $p$ in terms of the variable $q$ given an equation or relationship involving $p$ and $q$. 2. **Identify the relationship:*

1. We start with the given expression: $$(1+q)^2 - 4(1+q)(p+q)$$ 2. Expand each term:

1. The problem states: t is proportional to the square root of d, with t = 12 when d = 4. 2. To express t in terms of d, write $$t = k\sqrt{d}$$ for some constant k.

1. Stating the problem: Given that when $e^y$ is plotted against $x^3$, the graph is a straight line passing through points $(1, 13.5)$ and $(7.5, 0.5)$. 2. Find the equation of th

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

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