🧮 algebra

1. **Stating the problem:** We are given two quadratic functions and asked to analyze their properties including vertex, optimal value, axis of symmetry, zeros, y-intercept, direct

1. Énonçons le problème : Trouver le reste $r(X) = cX + d$ de la division du polynôme $P_m(X) = 2^m - 1^m - 1$ par $X^2 - 3X + 2 = (X-1)(X-2)$. 2. Rappel : Le reste d'une division

1. **State the problem:** Given roots $a$ and $b$ of an equation with conditions $a^2+b^2=21$ and $a+b=-3$, find the equation whose roots are $\frac{a}{b-2}$ and $\frac{b}{a-2}$. 2

1. **State the problem:** A store sold 150 apples in total, and 60 of these were green apples. We need to find what percentage of the apples sold were green. 2. **Formula used:** T

1. **State the problem:** Solve the cubic equation $$2x^3 + 2x^2 - 1040 = 0$$ for $x$. 2. **Write the equation:** $$2x^3 + 2x^2 - 1040 = 0$$

1. المشكلة: نريد معرفة أشكال معادلة الخط المستقيم المختلفة. 2. أشكال معادلة الخط المستقيم:

1. **State the problem:** Solve the equation $2^7 + 2^2 - 1340 = 0$. 2. **Recall the powers of 2:**

1. The problem involves understanding the expressions inside the logarithms and products: $$L_{\Lambda(1-\log_{\lambda}(1+\frac{\prod(\lambda^{1-\frac{\mu_1}{\Lambda}}-1)}{\lambda-

1. **State the problem:** Solve the equation $$e^2 + 5w = 12$$ for $w$. 2. **Identify the formula and rules:** This is a linear equation in $w$. To isolate $w$, subtract $e^2$ from

1. **Problem:** Solve the system $$\begin{cases} x \cos t - y \sin t = a \\ x \sin t + y \cos t = b \end{cases}$$

1. **State the problem:** We want to find what number divides the sum $4^{41} + 4^{42} + 4^{43}$ and check if it is divisible by 7. 2. **Rewrite the sum:** Factor out the smallest

1. **State the problem:** We want to find a number by which the sum $4^{41} + 4^{42} + 4^{43}$ is divisible. 2. **Rewrite the expression:** Notice that $4^{42} = 4^{41} \times 4$ a

1. Let's consider a simple algebra problem: Solve for $x$ in the equation $2x + 3 = 7$. 2. The formula used here is to isolate $x$ by performing inverse operations. We want to get

1. The problem is to find the solutions of an equation (not specified by the user). 2. Since the equation is not given, I cannot proceed with specific steps.

1. **State the problem:** Evaluate the expression $$\frac{3}{2-\sqrt{3}} - \frac{2}{3+\sqrt{3}}$$ without using a calculator. 2. **Rationalize the denominators:** To simplify expre

1. **State the problem:** Simplify or analyze the expression $4 \cdot 3x^2 - 10x + 8$. 2. **Rewrite the expression:** The expression is $4 \times 3x^2 - 10x + 8$.

1. **State the problem:** Evaluate $$\frac{3}{2}-\sqrt{3} - \left(\frac{2}{3}+\sqrt{3}\right)$$ without using a calculator and express the answer in the form $$m+n\sqrt{3}$$. 2. **

1. **State the problem:** Solve the inequality $$2x^2 + x - 1 > 0$$. 2. **Identify the type of inequality:** This is a quadratic inequality. To solve it, we first find the roots of

1. **State the problem:** Solve the linear equation $2x + 3y = 9$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation by mo

1. **State the problem:** Solve for $b$ given the equation $b^3 = 9 + 4\sqrt{5}$.\n\n2. **Recall the formula and approach:** We want to find $b$ such that when cubed, it equals $9

1. **Problem statement:** A train travels at one-third of its usual speed and arrives 30 minutes late. On the return journey, it travels at usual speed for 5 minutes, stops for 4 m