🧮 algebra

1. The problem is to prove that points lie on a line using the gradient (slope). 2. The formula for the gradient between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $$m = \frac{y_2

1. Énoncé du problème : Résoudre l'équation logarithmique $\log_2 (x + 1) = \log_2 5$. 2. Formule utilisée : Si $\log_a A = \log_a B$ alors $A = B$ (à condition que $a > 0$, $a \ne

1. **State the problem:** We want to simplify and understand the function $$y = \left(4 \ln(x^3)\right)^{\frac{1}{4}}$$. 2. **Recall the logarithm rule:** $$\ln(x^3) = 3 \ln(x)$$.

1. সমস্যাটি হলো: $a=3^{\frac{1}{3}}+3^{-\frac{1}{3}}$ হলে, $\frac{a^6-1}{a^3}$ এর মান নির্ণয় করতে হবে। 2. প্রথমে $a$ এর মানকে সহজভাবে প্রকাশ করি। এখানে $a = x + \frac{1}{x}$ যেখান

1. **State the problem:** Simplify the expression $$(b - 3b^2 + 7)(2 - 3b)(5 + b)$$. 2. **Recall the distributive property:** To multiply multiple polynomials, multiply two at a ti

1. **State the problem:** We want to find the least number of coins using 1¢, 5¢, 10¢, and 25¢ coins to make 143 cents (which is $1.43). 2. **Formula and approach:** Use the greedy

1. The problem is to generalize the equation $h = -5(t-2)^2 + 20$. 2. This equation is in vertex form of a quadratic function: $$h = a(t - h)^2 + k$$ where $(h, k)$ is the vertex a

1. **Énoncé du problème :** Montrer que $L = 6$ sachant que $$L = \frac{8}{3-\sqrt{5}} - \frac{\sqrt{40}}{\sqrt{2}}$$

1. The problem is to solve the equation or expression given by the user, but since no specific equation was provided, I will demonstrate solving a simple algebraic equation as an e

1. **State the problem:** Simplify the expression $$(2a+b)(2a-2b)$$. 2. **Use the distributive property (FOIL method):** Multiply each term in the first parenthesis by each term in

1. **State the problem:** Simplify the expression $$(a-b)(2a+b)$$. 2. **Recall the distributive property:** To multiply two binomials, multiply each term in the first binomial by e

1. **Énoncé du problème :** On considère la suite $(u_n)$ définie par $u_0 = 2$ et $u_{n+1} = (\sqrt{2} - 1) u_n + 2 - \sqrt{2}$ pour tout $n \in \mathbb{N}$.

1. **State the problem:** Simplify the complex fraction $$\frac{\frac{a-b}{a+b} - \frac{b}{a-b}}{1 + b \left(-\frac{2}{a+b} - \frac{3}{b-a}\right)}$$

1. Muammo: Toq funksiyalarni aniqlash. Toq funksiya uchun shart: $$f(-x) = -f(x)$$ har qanday $x$ uchun. 2. Berilgan funksiyalar:

1. **State the problem:** We need to analyze the function $$y = (x+1)^2 (x-4)^3$$. 2. **Formula and rules:** This is a polynomial function expressed as a product of powers of binom

1. **State the problem:** We are given the function $y = (x - 1)$ and need to understand or analyze it. 2. **Formula and explanation:** This is a linear function of the form $y = m

1. **Problem statement:** We want to find the number of years it takes for a country's population to double given an annual growth rate of $p\%$. 2. **Formula used:** The populatio

1. **Stating the problem:** Solve the equation $$\log_{\sqrt{27}} m = 2 \frac{2}{3}$$ for $m$. 2. **Recall the logarithm definition:** For $\log_a b = c$, it means $$a^c = b$$.

1. **State the problem:** We have a quadratic polynomial $$f(x) = x^2 + ax + b$$ and two conditions based on remainders when divided by linear factors: - When divided by $$x+1$$, t

1. The problem is to simplify the expression $$(x+h) [4(x+h)^2 + (x+h) -1] -4x^3 - x^2 + x$$ after expanding and distributing. 2. First, expand the terms inside the brackets:

1. **Problem Statement:** Find the H.C.F. (Highest Common Factor) and L.C.M. (Least Common Multiple) of the polynomials: $$x^3(x + 7) \quad \text{and} \quad x^4(x - 7)^2$$