🧮 algebra

1. **Stating the problem:** We want to verify the inequality and the recurrence relation given: $$U_{n+1} < \frac{2}{5} U_n \quad \forall n$$ 2. **Given expressions:** The user man

1. Let's clarify why we set $a-2=0$ in some problems. 2. Usually, this happens when we want to find the value of $a$ that makes an expression equal to zero or satisfies a certain c

1. **State the problem:** Given the quadratic function $f(x) = (a-2)x^2 + (b+3)x + (c-2)$, we are told it is zero degree and its range is 5. We need to find $a+b+c$. 2. **Understan

1. **State the problem:** Find the value of $k$ such that $\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2x}$.\n\n2. **Write the equation:** $$\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2

1. **State the problem:** Solve the equation $$2^{-2x+5} \times \frac{1}{4} = 2^{x+2}$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Note that $\frac{1}{4} =

1. **State the problem:** We are given two equations: $$P_A = \frac{30 + P_B}{3}$$

1. The problem is to simplify the expression $5p + 3 - 2p + 8$. 2. We use the rule that like terms (terms with the same variable) can be combined by adding or subtracting their coe

1. The problem is to understand and calculate combinations, often denoted as $nCr$, which represent the number of ways to choose $r$ elements from a set of $n$ elements without reg

1. The problem involves the ratio $5:7$ and some numbers $120$, $4$, and $3$, with variables $x$ and $y$. We need to find the values of $x$ and $y$ that satisfy the given ratio and

1. **Stating the problem:** We need to find the value of the expression $$\frac{6 + \sqrt{5}}{6 - \sqrt{5}} \times \frac{6 - \sqrt{5}}{6 + \sqrt{5}}$$.

1. The problem is to analyze the function $y = x^2 + 1$. 2. This is a quadratic function in the form $y = ax^2 + bx + c$ where $a=1$, $b=0$, and $c=1$.

1. **State the problem:** Find the slope of the line passing through the points $(-1,4)$ and $(4,-3)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1,y_1)$

1. Masalah: Buat tabel Charley untuk fungsi-fungsi yang diberikan yaitu $f_1(x) = 1 - x$, $f_2(x) = \frac{1}{x}$, $f_3(x) = \frac{1}{1 - x}$, $f_4(x) = \frac{x - 1}{x}$, $f_5(x) =

1. The problem asks to describe the relationship between distance traveled $D$ and time $t$ based on the graph. 2. The relationship is either proportional or not proportional. A pr

1. समस्या: हमें एक समीकरण हल करना है। 2. सबसे पहले, समीकरण को ध्यान से पढ़ें और समझें कि क्या पूछा गया है।

1. The problem states the relationship between pages printed $y$ and time $x$ in minutes is given by the formula $$y = 120x$$ where $y$ is the number of pages and $x$ is the time i

1. Problemi: Të kuptojmë nëse një funksion është funksion i përbërë. 2. Definicioni: Një funksion i përbërë është funksioni që formohet kur një funksion aplikohet mbi rezultatin e

1. **Problem:** Solve the system of equations: $$x + 4y = 5$$

1. The user asks to show a, b, c separately. 2. Since no specific problem or context is given, I will demonstrate how to represent variables a, b, and c separately.

1. Bài toán yêu cầu tìm các số nguyên tố $a$, $b$, $c$ sao cho thỏa mãn phương trình $$a^2 + 14b^2 + 2c^2 = 6abc.$$\n\n2. Đây là một phương trình đại số với ba ẩn nguyên tố. Ta sẽ

1. The problem is to simplify algebraic expressions, which means to rewrite them in a simpler or more compact form without changing their value. 2. The main rules to remember are: