🧮 algebra

1. **Problem statement:** A man starts at 0 on the number line and makes 101 moves. On the $n^{th}$ move, he moves $n$ units either left or right. We want to find how close to 0 hi

1. Given the equations: $$a^2 + b^2 = 90$$

1. **State the problem:** We are given the function $f(x) = (x - 1)^{\frac{5}{2}}$ and want to understand its behavior and graph. 2. **Analyze the function:** The function is defin

1. **State the problem:** We want to find the number of Pythagorean triples $(a,b,c)$ with positive integers where $a < b$ and $c = b + 2$, satisfying the Pythagorean equation $$a^

1. Stating the problem: Simplify the expression $$(8x-9b)(8x+9b)-x^2-(x-9b)^2$$. 2. Use the difference of squares formula for the first product: $$(8x-9b)(8x+9b) = (8x)^2 - (9b)^2

1. **Problem statement:** A man starts at 0 on the number line and makes 101 moves. On the $n$th move, he moves $n$ units either left or right. We want to find how close to 0 his f

1. The problem involves understanding the expressions $|x|^2$ and $|x^2| - 1$ and their relationship. 2. Recall that the absolute value squared of $x$ is $|x|^2 = (|x|)^2 = x^2$ be

1. **State the problem:** We are given a polynomial $$x^2 + ax + 1$$ and told it has only real zeros. We need to find the minimum value of $$a^2$$. 2. **Recall the condition for re

1. The problem asks us to list the numbers $\frac{1}{4}$, $\frac{4}{10}$, $\frac{41}{100}$, $0.04$, and $0.404$ from smallest to largest and find the middle number. 2. Convert all

1. **State the problem:** We need to find two positive integers whose sum is 60 and whose least common multiple (LCM) is 273. 2. **Recall the relationship between two numbers, thei

1. The problem is to determine the parity (odd or even) of a given number or expression. 2. A number is **even** if it is divisible by 2, i.e., it can be written as $2k$ where $k$

1. The problem asks to determine whether the number 0 is even. 2. By definition, an even number is any integer that can be expressed as $2k$ where $k$ is an integer.

1. **Problem statement:** Alexa created a 3x3 magic square where the sum of numbers in each row, column, and diagonal equals $K$. Some numbers are given, and we need to find the nu

1. Stating the problem: Simplify the expression $$-(x-2)^2-(x-5)(x+7)$$. 2. Expand the first term: $$-(x-2)^2 = -(x^2 - 4x + 4) = -x^2 + 4x - 4$$.

1. **State the problem:** We want to find how many integers between 100 and 999 have an odd ones digit, an even tens digit, and an even hundreds digit. 2. **Analyze the digits:**

1. The problem states: Let $$198 = 2 + 1a + 1b + 1c$$. We need to find the value of $$a + b + c$$. 2. Rewrite the equation clearly: $$198 = 2 + a + b + c$$.

1. Stating the problem: Simplify the expression $$6(x+5)^2 - 2(x-4)^2$$. 2. Expand each squared term:

1. **State the problem:** We need to find two positive integers whose sum is 60 and whose least common multiple (LCM) is 273. 2. **Recall the relationship between two numbers, thei

1. **Stating the problem:** We have a row of 3 squares made using 10 toothpicks. We need to find how many toothpicks are needed to make a row of 11 squares. 2. **Understanding the

1. The problem is to simplify the expression $$(2x+3)^2 - (9-4x)^2$$. 2. Recognize this as a difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$ where $$a = 2x+3$$ and $$b = 9-4x$$.

1. Stating the problem: Simplify the expression $$(x-6)^2 - (x-6)(x+6)$$. 2. Expand each term: