🧮 algebra

1. **Problem 1: Average height expression** The botanist has 4 plants with an average height of 23.2 cm. A fifth plant with height $x$ cm is added.

1. **Problemstellung:** Wir sollen die Gesamtkosten für die Herstellung eines Fußballs mit einem Durchmesser von 22 cm berechnen.

1. The problem asks us to describe the graph of the polynomial function $$p(x) = -2x^2 - x + 157$$ and identify its key features such as the direction it opens and its x-intercepts

1. **State the problem:** Solve the system of equations $$\begin{cases} -x + 4y = 6 \\ 2x - y = -5 \end{cases}$$

1. **State the problem:** We are given two functions $f(x) = x - 7$ and $g(x) = x^2 - 2x + 3$. We need to find the expression for the composition $g(f(x))$. 2. **Recall the composi

1. **State the problem:** Solve the system of equations: $$\begin{cases} 4x + 9y = 12 \\ -2x + y = -6 \end{cases}$$

1. **State the problem:** We have a cylindrical can with original radius $r = x$ cm and height $h = x$ cm.

1. **State the problem:** Determine if the point $(1,-2)$ satisfies the inequality $y \geq -6x - 3$. 2. **Substitute the point into the inequality:** Replace $x$ with $1$ and $y$ w

1. **State the problem:** We need to determine if the point $(3,4)$ satisfies the inequality $y < 4x - 2$. 2. **Write the inequality and substitute the point:** The inequality is $

1. **State the problem:** Solve the quadratic equation $$-\frac{\sqrt{7}}{2}x^2 - 7x - \sqrt{7} = 0$$. 2. **Identify coefficients:** The quadratic equation is in the form $$ax^2 +

1. **State the problem:** We are given two functions: \(f(x) = \left(\frac{3}{4}\right)^x\) and \(g(x)\) represented by a graph. We need to determine which statement about \(f(x)\)

1. **State the problem:** Solve the equation $$\left(a+\sqrt{2}\right)^2 + \left(\sqrt{2}a - 1\right)^2 = 6$$ for the variable $a$. 2. **Expand each squared term:**

1. **State the problem:** We want to find the year when the value of Sean's car, initially 20000, depreciates to approximately 5000 given a 19% annual depreciation rate. 2. **Formu

1. The problem asks to convert the logarithmic equation $\log 10 = 1$ into its equivalent exponential form. 2. Recall the definition of logarithm: $\log_b a = c$ means $b^c = a$.

1. **State the problem:** Solve the equation $$2\sqrt{7} + (x-2)(\sqrt{7} + x) = 0$$ for $x$. 2. **Expand the product:** Use the distributive property to expand $(x-2)(\sqrt{7} + x

1. **State the problem:** Evaluate the expression $\left(\log_3 27 + (5^2)^2\right) - \log_3 9$.\n\n2. **Recall logarithm rules and exponentiation:**\n- $\log_b a$ means the logari

1. **State the problem:** We are given that $P(x) = Q(x) \cdot (x - 3)$ and asked which of the statements A, B, C, or D must be true. 2. **Recall the property of multiplication:**

1. **State the problem:** We need to determine the end behavior of the function $f(x) = \ln(x - 3)$. 2. **Recall the domain and properties of logarithmic functions:** The natural l

1. **Problemstellung:** Berechne die Gesamtkosten für einen Fußball mit einem Durchmesser von 22 cm, wenn die Fixkosten pro Ball 5,63 und die zusätzlichen Materialkosten 78,35 pro

1. **Evaluate** $ (8+2)^2 \div 2(3) - (-5) $. Step 1: Calculate inside the parentheses: $8+2=10$.

1. **Problem:** Put the numbers $2.3 \times 10^2$, $\sqrt{400}$, $5^3$, $0.4$, and $1 \times 10^3$ in order from least to greatest. 2. **Step 1:** Calculate each number.