🧮 algebra

1. **State the problem:** Find the sum of the polynomials $$\left(\frac{1}{2} x^3 + 3x^2 - \frac{1}{4}\right) + \left(-\frac{1}{3} x^2 - 2x + 1\right).$$ 2. **Write the expression:

1. **Problem 1:** Determine the number of fire extinguishers needed for a building with 135,000 square feet, given that one extinguisher is needed for every 6,000 square feet. 2. *

1. **State the problem:** Luis needs to find the equation for the total cost $y$ to pour a concrete slab with area $x$ square feet, given an initial cost of 75 and an additional 2.

1. **State the problem:** Mr. Scarpa stays in a hotel that costs 78 per night. We need to write an equation for the total cost $c$ based on the number of nights $h$.

1. **State the problem:** We want to find the expression for the area of a large rectangle with dimensions $2x - 2$ (top side) and $x + 1$ (left side), from which a smaller rectang

1. The problem asks to complete statements about quadratic vocabulary. 2. The standard form of a quadratic equation is given by the formula:

1. **Problem Statement:** Find the axis of symmetry and vertex for the quadratic equation $y = x^2 + 6x + 4$. 2. **Formula for axis of symmetry:** The axis of symmetry for a quadra

1. **State the problem:** We have two rectangles: an outer rectangle with sides $5x + 8$ and $6x + 2$, and an inner rectangle with sides $x + 5$ and $3x$. The shaded region is the

1. **State the problem:** Solve the equation $$\frac{1}{x} = \frac{1}{5x} - \frac{x+3}{x^2}$$ for $x$. 2. **Identify the common denominator:** The denominators are $x$, $5x$, and $

1. **Problem:** Factor the expression $16n^2 - 9$ completely. 2. **Formula:** Recognize this as a difference of squares: $$a^2 - b^2 = (a + b)(a - b)$$

1. **Problem:** Factor the sum or difference of cubes for the expression $a^3 + 64$. 2. **Formula:** For sum of cubes, use $$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$$

1. **Problem:** Factor the expression $$16n^2 - 9$$ completely. 2. **Formula and rules:** This is a difference of squares, which factors as $$a^2 - b^2 = (a + b)(a - b)$$.

1. **State the problem:** A town has a population of 18000 and grows at 2% every year. We want to find the population after 11 years, rounded to the nearest whole number. 2. **Form

1. **State the problem:** We have a car purchased for 17900 dollars that depreciates at 12.25% per year. We want to find its value after 10 years.

1. **State the problem:** Simplify the expression $$\frac{\frac{3}{2a}}{\frac{4}{3a} - \frac{a}{4}}$$. 2. **Recall the formula:** To divide by a fraction, multiply by its reciproca

1. The problem asks to express the answer as coordinates in terms of $z$. 2. Typically, coordinates in terms of a variable $z$ can be written as $(x, y, z)$ where $x$ and $y$ are e

1. **State the problem:** Solve the system of linear equations for $x$, $y$, and $z$: $$\begin{cases}-3x + 2y - 3z = 4 \\ x + 2y + 3z = 1 \\ 2x + 4y + 6z = 2 \end{cases}$$

1. **State the problem:** Solve the equation $$6(y - 4) = -18$$ for $$y$$. 2. **Use the distributive property:** Multiply 6 by each term inside the parentheses.

1. **State the problem:** Solve the quadratic equation $-x^2 - 6x - 4 = 0$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $-1$ to simplify the leading coefficient.

1. **State the problem:** Solve the equation $4z^2 + 16z = 0$ for $z$. 2. **Formula and rules:** To solve quadratic equations, we can factor the expression and use the zero product

1. **State the problem:** Solve the equation $$12 - 8 = \frac{2}{3}x - 4$$ for $x$. 2. **Simplify the left side:**