🧮 algebra

1. **Problem:** Show that the square of any integer can be written in the form $4q$ or $4q+1$. 2. **Formula and rules:** The division algorithm states that for any integer $a$ and

1. **State the problem:** Solve the linear equation $$\frac{c}{2} - 4 = 6$$ for $c$. 2. **Write the formula and rules:** To solve for $c$, isolate $c$ on one side by undoing additi

1. **Statement of the problem:** Calculate the values of the piecewise function $$f(x) = \begin{cases} x+3 & \text{if } x \leq 0 \\ x^2 - 2 & \text{if } 0 < x \leq 5 \\ 2x + 4 & \t

1. **Énoncé du problème :** Calculer les valeurs de la fonction par parties $$f(x) = \begin{cases} x+3 & \text{si } x \leq 0 \\ x^2 - 2 & \text{si } 0 < x \leq 5 \\ 2x + 4 & \text{

1. **State the problem:** Simplify the surd $\sqrt[3]{56}$ completely. 2. **Recall the formula and rules:** To simplify cube roots, factor the number inside the root into prime fac

1. **State the problem:** Simplify the expression $$(a+b)^2 - 3(2b - a)$$. 2. **Recall formulas and rules:**

1. **State the problem:** Simplify the expression $5y + (y-2)(y+8)$. 2. **Recall the formula:** Use the distributive property (FOIL) to expand $(y-2)(y+8)$.

1. **State the problem:** Expand and simplify the expression $4u^3(2u^2 + 5u)$. 2. **Formula used:** Use the distributive property: $a(b + c) = ab + ac$.

1. **State the problem:** Solve the system of linear equations: $$\frac{1}{2}x + \frac{2}{3}y = 1$$

1. Let's start by stating the problem: solve the equation $-7a - 4 = -25$ for $a$. 2. The goal is to isolate $a$ on one side of the equation. To do this, we first get rid of the co

1. **State the problem:** A jet ski depreciates by 11% each year. The initial value is $8000. We need to find its value after 5 years. 2. **Formula used:** The value after deprecia

1. **State the problem:** Solve the equation $x^2 - x^3 = 12$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:

1. **Problem Statement:** Given two functions $f(x)$ and $g(x)$ with their graphs, we need to sketch the graph of $h(x) = f(x) - g(x)$.

1. **Problem 4:** Solve the system $$\begin{cases} 2x + y = -3 \\ 2x + 3y = 3 \end{cases}$$

1. **State the problem:** A tour boat travels 9 miles upstream and 9 miles downstream in a total of 4 hours. The river current speed is 3 miles per hour. We need to find the speed

1. **State the problem:** Simplify the expression $\left( \frac{2y^4}{4y} \right)^2$. 2. **Write the formula and rules:** When simplifying powers of fractions, use the rule $\left(

1. **State the problem:** Simplify the expression $$\left(\frac{zy^4}{4cy}\right)^2$$. 2. **Write the formula and rules:** When raising a fraction to a power, raise both numerator

1. **State the problem:** Rewrite the polynomial $x - x^5 - \frac{1}{3}$ in standard form. 2. **Recall the standard form of a polynomial:** A polynomial is written in standard form

1. **State the problem:** We are given the polynomial expression $$\frac{1}{6} + 3x$$ and need to identify its type, number of terms, constant term, leading term, and leading coeff

1. The problem is to add a number line representation to a math problem solution. 2. A number line is a visual tool to represent numbers as points on a straight line, helping to un

1. The problem is to understand and analyze the function $f(x) = \frac{1}{x^2}$. 2. This function is a rational function where the numerator is 1 and the denominator is $x^2$.