🧮 algebra

1. **State the problem:** Find the roots and the vertex of the quadratic function $$y = x^2 - 8x - 48$$. 2. **Formula for roots:** The roots are the solutions to $$y=0$$, so solve

1. Let's start with basic arithmetic operations involving positive and negative numbers. For example, consider the expression $-3 + 7 - 2$. 2. We perform the operations from left t

1. The problem asks for the domain of a function represented by a line segment on the x-axis. 2. The graph shows a line segment starting at $x = -3$ with an open circle, meaning $-

1. The provided text is a comprehensive Danish math workbook covering algebraic reduction, equations, and word problems. 2. It starts with basic arithmetic operations involving pos

1. **State the problem:** We are given the quadratic function $$y = x^2 - 8x - 48$$ and want to analyze its graph. 2. **Formula and rules:** A quadratic function is generally writt

1. **Problem statement:** Josh starts with a whole number (not 8 or 11) and counts by a whole number (not 5). Some of his numbers are 11, 32, and 46, but 24 is not one of his numbe

1. The problem is to find the domain of the function represented by the blue line segment on the coordinate plane. 2. The graph shows a line segment starting at the point $(-4,-4)$

1. **State the problem:** Find the roots and vertex of the quadratic equation $$y = x^2 + 10x - 11$$. 2. **Formula for roots:** The roots of a quadratic equation $$ax^2 + bx + c =

1. **State the problem:** Find the roots and vertex of the quadratic function $$y = -x^2 + 12x - 11$$. 2. **Formula for roots:** The roots of a quadratic $$ax^2 + bx + c = 0$$ are

1. **Problem:** Determine if $f(a) = (2a^2 + 3a + 1)^2$ is equivalent to $g(a) = 4(a^4 + 1.5a^3 + 3a^2 + 2a + 0.25)$. Justify your answer using algebra. 2. **Formula and rules:** T

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **State the problem:** Find the roots and vertex of the quadratic function $$y = -x^2 - 16x + 36$$. 2. **Formula for roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{

1. **Problem statement:** We will learn about "Reduktion og ligninger" which means "Reduction and equations" in Danish. This involves simplifying equations and solving for unknown

1. **State the problem:** A cashier scans 50 items in 10 minutes. We need to find how many minutes it takes to scan 32 items at the same rate. 2. **Formula and explanation:** The r

1. **State the problem:** The Pep Club made 168 pints of ice cream. Each container sold contains $\frac{3}{4}$ pint. We need to find how many containers they sold and how much mone

1. Il problema chiede se un calcolo è corretto, ma non è stato fornito alcun calcolo specifico. 2. Per poter verificare un calcolo, è necessario conoscere l'espressione o l'equazio

1. **Problema:** Calcolare la data media ponderata dei tre periodi di indennità, tenendo conto della durata e degli importi di ciascun periodo. 2. **Formula e regole:** La data med

1. **State the problem:** Simplify the expression $$\left(5^8 \cdot 5^4 \cdot 5 \div 5^{11}\right) \cdot 3^0 - \left(15^6 \cdot 15 \div 15^3\right) \div \left(\frac{30}{2}\right)^3

1. **Problem statement:** Find the leading coefficient $a$ of a fifth-degree polynomial with rational coefficients given zeros at $-3$, $4$ (multiplicity 2), and $1 + \sqrt{5}$, an

1. **State the problem:** We want to solve the equation $$\frac{2x+4}{3} = 5$$ for $x$. 2. **Formula and rules:** To solve equations involving fractions, multiply both sides by the

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