🧮 algebra

1. **Problem Statement:** Given two positive integers $p$ and $q$ expressed as $p = a b^2$ and $q = a^3 b$, where $a$ and $b$ are prime numbers, find the least common multiple (LCM

1. **State the problem:** Factorize the cubic polynomial $$3x^3 + 26x^2 + 61x + 30$$. 2. **Recall the method:** To factor a cubic polynomial, we try to find rational roots using th

1. **State the problem:** We need to find the length and width of a rectangle where the area is 55 yd² and the length is 4 yd less than three times the width. 2. **Write the formul

1. **Stating the problem:** We want to find the value of $a$ such that the expression $M = 2^3 \times 3^2 \times a$ is a perfect cube. 2. **Recall the rule for perfect cubes:** A n

1. **State the problem:** Solve the equation $$\log_3(3x^2) + \log_3(3) = 2$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms with the same base can be combine

1. **State the problem:** Solve the logarithmic equation $$\log_2 x + \log_2 (x - 2) = 3$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms with the same base c

1. **State the problem:** Solve the equation $$\log_2(x^2 - x - 6) = 2$$ and find the exact solutions and their approximations to the nearest tenth. 2. **Recall the logarithm defin

1. **State the problem:** We need to sketch the graph of the function $$f(x) = 3x^4 - 4x^3$$. 2. **Understand the function:** This is a polynomial function of degree 4. The general

1. **Problem statement:** We need to find three consecutive odd numbers whose sum is 69, and then identify the third number in the sequence. 2. **Define variables:** Let the first

1. **State the problem:** We need to find the sixth number in a sequence of 6 consecutive integers whose sum is 387. 2. **Set up variables:** Let the first integer be $x$. Then the

1. **State the problem:** We need to find the first of 4 consecutive even numbers whose sum is 36. 2. **Define variables:** Let the first even number be $x$. Then the next three co

1. **Problem statement:** We are given that the sum of 6 consecutive odd numbers is 204, and we need to find the fourth number in this sequence. 2. **Define variables:** Let the fi

1. **State the problem:** We have two numbers, one is positive and is 3 times the other. If we add 2 to the larger number and 5 to the smaller number, then one becomes twice the ot

1. **State the problem:** We need to solve the system of inequalities by graphing: $$y \leq 1$$

1. The problem asks us to find the value represented by the symbol \tealD6 in the number $$97\tealD6.14$$. 2. Since \tealD6 is used in a number with digits before and after the dec

1. **State the problem:** We need to graph the rational function $$f(x) = \frac{-x + 5}{-2x + 1}$$ and identify its vertical and horizontal asymptotes. 2. **Find vertical asymptote

1. **Problem Statement:** Given a rational function $f$ with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=2$, find the equations of the asymptotes, the $x$- and $

1. Let's consider a simple algebra problem: Solve for $x$ in the equation $$2x + 3 = 7$$. 2. The formula to solve for $x$ in a linear equation $ax + b = c$ is to isolate $x$ by sub

1. **Problem Statement:** Create an algebraic expression with at least three unlike terms and simplify it. 2. **Step 1: Write an expression with unlike terms.**

1. El problema pregunta si para una matriz invertible $A$, la matriz $A + A^{-1}$ siempre es invertible. 2. Recordemos que una matriz $A$ es invertible si existe $A^{-1}$ tal que $

1. Planteamos el problema: Tenemos dos parábolas, $P_1$ y $P_2$. La parábola $P_2$ tiene vértice en el origen $(0,0)$ y pasa por el punto $(2,1)$. La parábola $P_1$ pasa por los pu