🧮 algebra

1. **Problem (a): Simplify \(\frac{x^2 + 7x}{x^2 - 49}\)** 2. First, factor both numerator and denominator:

1. **பிரச்சினையை விளக்குதல்:** $r \in \mathbb{Z}^+$ என்றால், $U_r = \frac{1}{r(r+2)} \cdot \frac{1}{r+4}$ மற்றும் $f(r) = \frac{1}{r(r+2)}$ என கொடுக்கப்பட்டுள்ளது.

1. Let's clarify the problem: You are asking why we divide an expression by $4x$ when the expression contains $4y$. 2. Typically, division by $4x$ is done to isolate a variable or

1. **State the problem:** We need to find the value of $\frac{y}{x}$ given the equation $9^x = 81^y$. 2. **Rewrite the bases as powers of the same base:**

1. **State the problem:** You asked, "From where did the 9 come from?" in the expression $\frac{9}{x}$. Let's clarify this. 2. **Recall the original problem:** The original equatio

1. **State the problem:** Given the equation $9^x = 81^y$, find the value of $\frac{9}{x}$. 2. **Rewrite the bases as powers of 3:**

1. **State the problem:** Solve for $x$ in the equation $$16(3)^x = 81(2)^x.$$\n\n2. **Rewrite constants as powers:** Note that $16 = 2^4$ and $81 = 3^4$. Substitute these into the

1. **State the problem:** Solve for $x$ in the equation $16(3)^2=81(2^x)$. 2. **Write down the equation:**

1. **State the problem:** Solve for $x$ in the equation $16(3)^2 = 81(2^{2x})$. 2. **Write down the equation:**

1. The problem is to determine which filter design meets both requirements: - Remove at least 95% of chlorine from tap water.

1. **Problem:** Solve the equation $$6x^4 - 11x^2 + 3 = 0$$ 2. **Formula and approach:** This is a quartic equation but can be treated as a quadratic in terms of $x^2$. Let $y = x^

1. **State the problem:** Solve the system of linear equations: $$3x - 6y = -2$$

1. **State the problem:** Solve the equation $$x \times \ln(x) - \frac{7}{4} \times (\ln(x))^2 - x + 1 = 0.$$\n\n2. **Rewrite the equation:** Group terms to see if substitution hel

1. **Problem Statement:** Given the sequence $U_r = \frac{4(2r+7)}{(2r+1)(2r+3)(2r+5)}$ for $r \in \mathbb{Z}^+$, and a function $f(r) = \frac{A}{2r+1} + \frac{B}{2r+3}$ where $A,B

1. **State the problem:** Solve the equation $x \times \ln(x) = 0$ for $x$. 2. **Recall the zero product property:** If a product of two factors equals zero, then at least one of t

1. সমস্যা: একটি সরলরৈখিক সমীকরণ সমাধান করুন: $$3x + 5 = 20$$ 2. সূত্র: সরলরৈখিক সমীকরণের সমাধানে আমরা লক্ষ্য করি যে, সমীকরণের দুই পাশে সমান অপারেশন করলে সমীকরণের সমতা বজায় থাকে।

1. **Problem statement:** Simplify and prove that the expression $$A = \left( \frac{x - 2}{x^3 - 3x^2} - \frac{6}{x^4 - 9x^2} + \frac{x + 2}{x^2 + 3x^2} \right) \cdot \frac{4042}{x

1. **State the problem:** We are given that 38% of the students are girls and the number of boys is 1023. We need to find the total strength of the school. 2. **Define variables:**

1. Problema: Resolver a equação $$\sqrt[3]{2x(x-6)} = 0$$. 2. Fórmula e regra: A raiz cúbica de um número é zero se e somente se o número dentro da raiz cúbica for zero. Ou seja, $

1. **Problem 1a:** Write the equation of the circle with center at (-2, 3) passing through (3, 7). 2. The formula for a circle with center $(h,k)$ and radius $r$ is:

1. **State the problem:** Solve for $u$ in an equation involving $u$ (the exact equation is not provided, so let's assume a general linear equation for demonstration: $au + b = c$.