🧮 algebra

1. **State the problem:** We are given a rectangle divided into four smaller rectangles with terms along the sides: $7z$ and $-4$ on one side, and $7z$ and $4$ on the other side. W

1. **State the problem:** Find the product of the binomials $$(7z + 4)(7z - 4)$$. 2. **Formula used:** This is a difference of squares pattern, where $$(a + b)(a - b) = a^2 - b^2$$

1. **State the problem:** We need to solve the expression $\frac{1}{5} + \left(-\frac{4}{5}\right)$. 2. **Recall the rule for adding fractions with the same denominator:** When add

1. **State the problem:** We are given the function $y = 3(x + 5)(x - 2)$ and want to understand its properties. 2. **Formula and rules:** This is a quadratic function in factored

1. **State the problem:** We are given a triangle with sides labeled $a^2 - 10$, $a^2 - 5a$, and $6$. We want to find the value(s) of $a$ that satisfy the triangle's conditions. 2.

1. Problem b: Simplify $5^{\frac{1}{4}} \cdot 5^{\frac{11}{4}}$. 2. Use the rule of exponents: $a^m \cdot a^n = a^{m+n}$.

1. **State the problem:** Simplify the expression $\left(4\sqrt{5}-3\right)\left(\sqrt{5}-2\right)$.\n\n2. **Use the distributive property (FOIL method):** Multiply each term in th

1. **State the problem:** A gopher runs from the southeast corner of a rectangular yard to the opposite corner by running the width and then the length, each 6.1 meters, and then r

1. The problem asks for the expression representing the perimeter of a rectangle with width $(10f + 8)$ cm and length $(5f - 3)$ cm. 2. Recall the formula for the perimeter $P$ of

1. **State the problem:** Solve the equation $$\frac{2x+4}{x} + \frac{x+2}{x-2} = \frac{3x}{x-2}$$ for $x$. 2. **Identify the domain restrictions:** The denominators cannot be zero

1. **State the problem:** Simplify the expression $$\frac{3}{x-3} - \frac{5}{x+2}$$. 2. **Find a common denominator:** The denominators are $x-3$ and $x+2$. The common denominator

1. **State the problem:** We are given the function $$h(x) = (-x + 2)^2$$ and the parent function $$f(x) = x^2$$. We need to identify the transformations applied to $$f(x)$$ to get

1. **State the problem:** We are given the function $h(x) = (x - 4)^2$ and the parent function $f(x) = x^2$. We need to identify the transformations applied to $f(x)$ to get $h(x)$

1. The problem asks to identify which quadratic functions have specific domain and range properties based on their graphs. 2. Recall that the domain of a function is the set of all

1. **State the problem:** We need to find the range of each quadratic function given either by formula or table of values. 2. **Recall the general form and range of a quadratic fun

1. **Stating the problem:** Calculate the total pay for an employee who works 35 regular hours and 5 overtime hours, given the regular hourly rate and overtime multiplier. 2. **Giv

1. **Problem Statement:** Calculate the gross pay for a sales job with a base salary and commission. 2. **Given:** Base salary = 350 per week, Commission rate = 8% = 0.08, Sales =

1. **State the problem:** We need to find the positive solution to the quadratic equation $$7x^2 - 20x - 32 = 0.$$\n\n2. **Recall the quadratic formula:** For an equation $$ax^2 +

1. The problem is to find one solution to the equation $$3x(x - 4)(x + 5) = 0$$. 2. The zero product property states that if a product of factors equals zero, then at least one of

1. **State the problem:** We are given two functions: the original cubic function $f(x) = x^3$ and a transformed function $h(x) = -(x + 2)^3 - 4$. We want to understand the transfo

1. The problem asks to describe the transformation from the function $f(x) = x^3$ to $g(x) = (x + 4)^3$. 2. Recall that for a function $f(x)$, the transformation $f(x + h)$ shifts