🧮 algebra

1. **Stating the problem:** We are given a sequence with terms corresponding to positions $n=100, 101, 102$ having values $412, 416, 420$ respectively. We want to find the general

1. **State the problem:** We are given the quadratic sequence: -5, -2, 3, 10, 19, ... and need to find the formula for the $n^{th}$ term. 2. **Recall the formula for a quadratic se

1. **State the problem:** We are given the quadratic sequence: -5, -2, 3, 10, 19, ... and need to find the formula for the $n^{th}$ term. 2. **Identify the pattern:** Quadratic seq

1. **State the problem:** We are given the equation of two lines combined as $$x^2(\tan^2\theta + \cos^2\theta) - 2xy \tan\theta + y^2 \sin^2\theta = 0$$ and need to show that the

1. **Problem statement:** Show that $x+2$ is a factor of the polynomial $p(x)$. 2. **Key concept:** A polynomial $p(x)$ has a factor $x - a$ if and only if $p(a) = 0$. This is know

1. **State the problem:** We need to find the remainder when the polynomial $p(x) = 15x^3 + 22x^2 - 15x + 2$ is divided by $x + 1$. 2. **Recall the Remainder Theorem:** The remaind

1. **State the problem:** We are given a cubic polynomial $p(x) = 2x^3 + ax^2 + bx + c$ with roots $x=\frac{1}{2}$, $x=n$, and $x=-n$, where $a,b,c,n$ are integers. The $y$-interce

1. **State the problem:** Simplify the expression $\left(3x^{-2} y\right)^2$. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to tha

1. **Problem statement:** There are 25 students who took math and science exams. 17 passed science, 8 passed both math and science, and 3 did not pass either exam. We need to find

1. **Problem:** Simplify $9^{3/2}$. 2. **Formula and rules:** For any positive number $a$ and rational exponent $m/n$, we use the rule:

1. समस्या: भिन्नों को मिश्र भिन्न में बदलना। 2. मिश्र भिन्न में बदलने का सूत्र: यदि भिन्न $$\frac{a}{b}$$ है, जहाँ $$a > b$$, तो इसे मिश्र भिन्न के रूप में लिखा जाता है:

1. **Problem Statement:** Given the total cost of 3 mangoes and 4 apples is 160, and after increasing the cost of a mango by 20% and decreasing the cost of an apple by 20%, the tot

1. **State the problem:** We need to find the points of intersection A and B between the line $y = 5x + 6$ and the curve $xy = 8$. 2. **Write down the equations:**

1. **State the problem:** We need to find the equation of a water pipe line whose gradient is the reciprocal of a given prime number. The line must pass through the point $P(a,b)$,

1. **State the problem:** Marcy burns 80% of the calories walking compared to jogging. We want to write a function for calories burned walking as a function of calories burned jogg

1. The problem asks whether the left-hand side (LHS) of an equation or expression is positive or negative at $x=0$. 2. To determine this, we need the explicit form of the LHS funct

1. The problem asks to write the left-hand side (LHS) of an inequality evaluated at a specific point $x_0$. 2. Generally, an inequality can be written as $f(x) \leq g(x)$ or $f(x)

1. **State the problem:** We want to solve the inequality $$(x-3)(x^2+1)(x^2+2x-15) > 0.$$ 2. **Find the zeros (roots) of the polynomial:** The zeros occur where each factor equals

1. **State the problem:** A car rental agency charges 20 per day to rent a car. We need to write the cost as a function of the number of days rented and find the cost for 5 days. 2

1. **State the problem:** Calculate how much a waitress makes for a 4-hour and an 8-hour shift given the pay function $P(t) = 2.52t + 78$, where $t$ is the number of hours worked.

1. **Problem:** Simplify the expression $12 + 14\frac{1}{2} + \frac{1}{4}21 + 41$. 2. **Step 1:** Convert mixed numbers and fractions to improper fractions or decimals for easier a