🧮 algebra

1. **Expand and simplify:** Given $f(x) = -3(x - 2)(x + 4)$ Use the distributive property (FOIL) to expand:

1. **State the problem:** Ambaye has cows and oxes. Together, they have 512 eyes and 1024 legs. We know Ambaye has 204 oxes. We need to find how many cows he has. 2. **Define varia

1. **State the problem:** You have a test with geography problems worth 5 points each and history problems worth 15 points each.

1. **State the problem:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can use f

1. **Problem statement:** We have two sequences $(U_n)$ defined by recurrence relations and related sequences $(V_n)$ defined in terms of $(U_n)$. We want to show that $(V_n)$ defi

1. **Problem statement:** Consider the sequence $(U_n)$ defined by $U_0=1$ and $U_{n+1} = \frac{9}{6 - U_n}$ for all $n \in \mathbb{N}$. Show that $0 < U_n < 3$ for all $n$. 2. **F

1. The problem asks to simplify sums and differences of decimal numbers, some with repeating decimals indicated by an overline. 2. To add or subtract repeating decimals, convert ea

1. مسئله: دامنهٔ تعریف تابع $$f(x) = \sqrt{\frac{x}{x^2}} - \frac{r}{r} + \sqrt{\frac{2x}{r}} - x r$$ را پیدا کنید. 2. ابتدا باید شرایطی که زیر رادیکال‌ها منفی نشوند و مخرج‌ها صفر

1. **State the problem:** Make $w$ the subject of the equation $$2h = \frac{1}{5}fw$$. 2. **Understand the formula:** The equation relates $h$, $f$, and $w$. We want to isolate $w$

1. **Problem Statement:** Find the value of $x$ when $y = -15.5$ for the line of best fit from the first data set:

1. **State the problem:** Multiply $$4 \times \sqrt[3]{4x^2} \times \left(2\sqrt[3]{32x^2} - x\sqrt[3]{2x}\right)$$ using the distributive property. 2. **Recall the distributive pr

1. The problem is to find the x-intercept(s) of a polynomial function. 2. The x-intercept(s) are the points where the graph of the polynomial crosses the x-axis.

1. **State the problem:** We have a parabola representing the ball's height: $$y = -x^2 + 6x$$ and a horizontal line at height $$y = 8$$. We want to find where the ball reaches thi

1. **State the problem:** We have a rectangle with area given by $A = x^2 + 5x$ square meters. 2. **Part (a): Find the dimensions if the area is 24 m².**

1. **State the problem:** Find the x- and y-intercepts of the parabola given by the equation $$y = 4x^2 - 12x + 9$$. 2. **Recall the definitions:**

1. The problem is to solve the inequality $$\frac{(x-1)(x+3)^2}{(x-4)} \leq 0$$ and determine how many solutions satisfy it. 2. Important points to consider are the zeros of the nu

1. **State the problem:** Solve the equation $64^{2x-1} = 128^x (2^{x-1})$ for $x$. 2. **Rewrite bases as powers of 2:**

1. **Problem statement:** Find the distance between the points $(2,7)$ and $(x^2, x-1)$ rounded to 3 decimals. 2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and

1. Énonçons le problème : Simplifier et comprendre l'expression $\frac{1}{x}(x + \ln x)$.\n\n2. Rappelons la propriété distributive : $a(b + c) = ab + ac$. Ici, $\frac{1}{x}$ est m

1. **Problem:** Determine which statement is true for the power function $f(x) = x^n$ where $n$ is a negative even integer. 2. **Recall:** A power function $f(x) = x^n$ with $n$ ne

1. The problem is to redefine the modulus function $F(x) = |x^2 - 8x + 15|$ without the absolute value. 2. First, factor the quadratic inside the absolute value: