🧮 algebra

1. **Problem statement:** Berechne, um wie viel Prozent der Preis verändert wurde bei zwei aufeinanderfolgenden Preisänderungen und wie viel Prozent die gesamte Preisänderung beträ

1. **Stating the problem:** Simplify the expression $$\frac{x^4 - x^2}{5x + 5} : \frac{x - 1}{10x}$$.

1. **State the problem:** Solve the equation $\log_2 x = \log_4 (x+6)$.\n\n2. **Recall the change of base formula and properties of logarithms:**\nWe know that $\log_a b = \frac{\l

1. The problem is to solve the equation $$\frac{2x+3}{4} = 5$$ for $x$. 2. The formula used here is to isolate $x$ by eliminating the denominator and then solving the resulting lin

1. **State the problem:** We are given a temperature model for coffee cooling: $$T(t) = 56e^{kt} + 22$$

1. **State the problem:** We have a coffee temperature function $$f(t) = 56e^{kt} + N$$ where $$f(t)$$ is the temperature at time $$t$$ minutes, and constants $$k$$ and $$N$$ need

1. **State the problem:** We have the function $f(x) = x^3$ and want to analyze the effect of reflecting its graph across the x-axis and then the y-axis. 2. **Reflection across the

1. **Classify each function as odd, even, or neither.** We use the definitions:

1. **State the problem:** Factor the cubic polynomial $$a^3 - 3a + 2$$ completely. 2. **Recall the formula and rules:** To factor a cubic polynomial, we can try to find rational ro

1. **State the problem:** We have a sequence starting at 11, and each term is found by subtracting 4 from the previous term. 2. **Formula used:** The nth term of an arithmetic sequ

1. **State the problem:** Find the value of $a^2$ when $a = 6$. 2. **Formula used:** The expression $a^2$ means $a$ multiplied by itself, so

1. **Problem (a):** Find the vertex, focus, and directrix of the parabola given by $$(y - 7)^2 = 8(x - 2)$$ 2. **Formula and rules:** This is a parabola that opens horizontally bec

1. The problem asks us to determine whether the quadratic function $$f(x) = -2(x + 8)^2 - 5$$ has a minimum or maximum, and then find that minimum or maximum value. 2. The general

1. The problem is to identify the equation that best matches the given graph of a parabola. 2. The graph shows a parabola with vertex at approximately $(-3, 3)$ and it opens downwa

1. The problem asks to determine the axis of symmetry for a parabola that opens upward and has its vertex at (0, -4). 2. The axis of symmetry of a parabola is a vertical line that

1. The problem asks for the x-intercepts of the quadratic graph. 2. X-intercepts occur where the graph crosses the x-axis, meaning the y-value is zero.

1. **State the problem:** We need to find which ratios are equivalent to $1:4$. 2. **Recall the rule for equivalent ratios:** Two ratios $a:b$ and $c:d$ are equivalent if $\frac{a}

1. **State the problem:** We need to find which ratios are equivalent to 15:9. 2. **Formula and rule:** Two ratios $a:b$ and $c:d$ are equivalent if $\frac{a}{b} = \frac{c}{d}$.

1. The problem asks us to find which ratios are equivalent to $3:9$. 2. Two ratios $a:b$ and $c:d$ are equivalent if $\frac{a}{b} = \frac{c}{d}$.

1. **State the problem:** We have 3 cups of cereal and 2 cups of cheese puffs in a snack mix. 2. **Part-to-part ratio of cheese puffs to cereal:** This ratio compares the amount of

1. **State the problem:** We have 7 cups of kiwis and 3 cups of bananas in a fruit salad. 2. **Part-to-part ratio of bananas to kiwis:** This ratio compares the amount of bananas t