🧮 algebra

1. Das Problem lautet: Löse die Gleichung $2x + 3 = 7$ nach $x$ auf. 2. Die verwendete Formel ist die lineare Gleichung $ax + b = c$, wobei $a$, $b$ und $c$ Konstanten sind.

1. **Stating the problem:** We have a grid of equations with numbers 2, 3, 9, and empty circles connected by plus and equals signs. The goal is to find the missing numbers in the e

1. **State the problem:** We are given the quadratic function $f(x) = -2x^2 + 3x - 25$ and need to analyze it. 2. **Find the domain:** The domain of any quadratic function is all r

1. The problem is to evaluate $\log_3\left(\frac{81}{27}\right)$.\n\n2. Recall the logarithm rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.\n\n3. Apply the rule: $

1. **State the problem:** Expand and simplify the expression $$(a - 4x)\left(\frac{1}{2}a + 3x\right)$$. 2. **Formula used:** Use the distributive property (FOIL method) to multipl

1. **State the problem:** Simplify the expression $\frac{3}{4} - \frac{1}{4} + \frac{5}{4} \times \frac{2}{5}$.\n\n2. **Recall the order of operations:** Multiplication comes befor

1. **Problem a:** Simplify $\left(\frac{a}{b}\right)^{\log 0.5} \cdot \left(\frac{a}{b}\right)^{\log 0.2}$. 2. Use the property of exponents: $x^m \cdot x^n = x^{m+n}$.

1. **State the problem:** Simplify the expression $$\frac{7}{10} - \frac{3}{5} \times \frac{1}{2} + \frac{3}{4}$$. 2. **Recall order of operations:** Multiplication comes before ad

1. **State the problem:** We are given a rectangle with an area of 15 cm². The height is 5 cm and the width is given as $q - 2$ cm. We need to find the value of $q$. 2. **Formula f

1. **State the problem:** Solve the equation $f(|x|) = 0$ where $f(x) = 3|x - 2| - 10$. 2. **Write the equation:**

1. **State the problem:** We want to show mathematically that the line $y=5x+10$ intersects only one branch of the graph of $f(x)=3|x-2|-10$. 2. **Recall the definition of $f(x)$:*

1. **State the problem:** We have a quadratic function $y = x^2 + ax + b$ where $a,b \in \mathbb{Z}$. Points $P(-1,-2)$ and $Q(3,2)$ lie on the curve. We need to find two equations

1. **State the problem:** We have chickens and cows on a farm. Chickens have 2 legs each, cows have 4 legs each, and the total number of legs is 82. 2. **Define variables:** Let $x

1. **State the problem:** We are given the equation of a line $y = -8x + 8$ and a table showing possible combinations of chickens ($x$) and cows ($y$) on a farm, where the total nu

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{x-3} = 3$$ for $x$. 2. **Recall the formula and rules:** To solve a rational equation like this, multiply both

1. The problem is to graph the function $y = \frac{1}{x} + 2$ by starting with the graph of the standard function $y = \frac{1}{x}$ and applying transformations. 2. The standard fu

1. **State the problem:** Solve the equation $$\frac{2}{3}x = 10$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ by dividing both sides of the equatio

1. The problem is to analyze the function $$y = \frac{1}{x} + 2$$. 2. This function is a rational function with a vertical asymptote where the denominator is zero, i.e., at $$x=0$$

1. **State the problem:** Solve the equation $$\frac{1}{3}x - \frac{2}{3} = \frac{1}{4}x$$. 2. **Identify the least common multiple (LCM) of denominators:** The denominators are 3

1. The problem asks to analyze the function $h(d) = - \frac{1}{9} (d - 4)^2 + 4$ which models the height of a basketball ball as a function of horizontal distance $d$. 2. This is a

1. The problem gives two functions: $$h(d) = - \frac{1}{9} (d - 4)^2 + 4$$ and $$h(d) = - \frac{1}{9} (d + 1)^2 + 4$$. We are asked to analyze these parabolas, which model the heig