🧮 algebra

1. **State the problem:** Mark's car gets 30 miles per gallon and Jodi's car gets 21 miles per gallon. Both used a whole number of gallons to travel from their home to the theater.

1. The problem is to solve the system of equations. However, the system is not provided in the message, so I cannot solve it without the equations. Please provide the system of equ

1. **Stating the problem:** We are given the function $$f(x) = \frac{1}{6}x^3 + \frac{3}{2}ax^2 - 36a^2x + 10$$ where $$a \in \mathbb{R}^+$$ (positive real numbers).

1. **Problem:** Find the roots of the polynomial $f(x) = x^4 - 10x^3 + 20x^2 + 19x - 12$. 2. **Step 1: Use synthetic division to factor the polynomial.**

1. **Problem statement:** Löse die Gleichung $$16 = ?$$

1. **問題の説明** 与えられた和 $$\sum_{k=1}^n (-x)^{k-1}$$ を $x, n$ の式で表す。

1. **State the problem:** Given $p=100000$ and the equation $p^2 + 2pq + q^2 = 1$, find the value of $q$. 2. **Recall the formula:** The equation $p^2 + 2pq + q^2 = 1$ can be recog

1. **State the problem:** Solve the equation $$(-4) + (-1) + 2 + 5 + \ldots + x = 437$$ where the terms form an arithmetic sequence. 2. **Identify the sequence:** The terms are $$-

1. The problem asks if $\log(7/5)$ divided by $\log(3/4)$ is the same as $\lg(7/5)/\lg(3/4)$. 2. Here, $\lg$ typically denotes the logarithm base 10, and $\log$ without a base ofte

1. **State the problem:** Simplify the expression $$5^{3 - \log_2 25} + 3^{2 - \log_3 3} - 4^{4 - \log_2 5}.$$\n\n2. **Recall logarithm and exponent rules:**\n- $a^{m-n} = \frac{a^

1. **State the problem:** Solve the equation $5 \cdot 6^x = 7 \cdot 8^x$ for $x$. 2. **Rewrite the equation:** We want to isolate $x$. Start by dividing both sides by $7 \cdot 6^x$

1. Het probleem is om gelijknamige breuken te rangschikken, dat wil zeggen breuken met dezelfde noemer. 2. Bij gelijknamige breuken vergelijken we alleen de tellers, omdat de noeme

1. **State the problem:** We want to understand how to go from the equation $$(\ln 3)(\ln 3 + \ln x) = (\ln 4)(\ln 4 + \ln x)$$

1. **Problem statement:** Löse die Gleichung $$ (x - 29)(-2x + 14) = 0 $$ und finde die Werte von $x_1$ und $x_2$. 2. **Formel und Regel:** Wenn das Produkt zweier Faktoren null is

1. Enunciado: Determine o valor de $$\log_a \left( \frac{a^2}{\sqrt[3]{b}} \right) \times \log_b \left( \frac{c^2 \sqrt{b}}{a^3} \right)$$

1. Das Problem scheint darin zu bestehen, verschiedene Werte bei bestimmten Nummern (5, 6, 7) zu vergleichen, möglicherweise aus einer Berechnung oder Messung. 2. Wir betrachten di

1. **State the problem:** Solve the equation $$|p-1| - |2-p| = -10$$. 2. **Recall the definition of absolute value:** For any real number $x$, $$|x| = \begin{cases} x & \text{if }

1. **Problem statement:** Simplify the expression $-4^5 + 11 \cdot 4^5$. 2. **Recall the distributive property:** For terms with the same base and exponent, $a \cdot x^n + b \cdot

1. The problem is to find the value of $x$. 2. Since no equation or context is given, we cannot solve for $x$ directly.

1. **State the problem:** Mrs. Mary Moolah invested a total of $20,000 in two types of bonds. One bond pays 5% interest, and the other pays 8%. The total interest earned from both

1. **State the problem:** Solve the equation $$\frac{5}{2x + 1} = \frac{3}{x + 3}$$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products