🧮 algebra

1. **State the problem:** We want to analyze the function $$y=\frac{1}{\sqrt{x^2+4x+3}}$$ and understand its behavior. 2. **Rewrite the expression inside the square root:** Factor

1. **Problem:** In an arithmetic progression (AP), the first term $a_1=2$ and the sum of the 1st and 6th terms is $16 \frac{1}{2} = 16.5$. Find the 4th term $a_4$. 2. **Formula and

1. **State the problem:** We have the equation $$3|x - 8| = k$$ where $$k$$ is a constant. We want to find the value(s) of $$\frac{k}{3}$$ such that the equation has exactly one so

1. The problem is to write the expression $4x3 \times \frac{2}{-3} - (-1) \times 3 \times (-3) \times 2$ in proper mathematical format. 2. First, interpret the expression carefully

1. Problem: Multiply $\frac{5y^3}{32x}$ and $\frac{-4}{15x^2 y^3}$ and simplify. Step 1: Multiply numerators and denominators:

1. **State the problem:** Simplify the expression $4 \times 3 \times \frac{2}{-3} - (-1) \times 3 \times (-3) \times 2$. 2. **Recall the rules:**

1. **State the problem:** Solve the equation $0.10x + 0.20y = 0.19(x + y)$ for $y$ in terms of $x$. 2. **Write the equation:**

1. **Problem:** Multiply the rational expressions $\frac{5y^3}{32x} \times \frac{-4}{15x^2 y^3}$ and simplify. 2. **Step 1: Factor all numerators and denominators completely.**

1. **State the problem:** Simplify the expression $5y^3 \cdot 32x - 4 \cdot 15x^3y^3$. 2. **Rewrite the expression clearly:**

1. **Problem:** Solve the equation $k(2k + 5) = 3$. 2. **Formula and rules:** This is a quadratic equation. Expand and rearrange to standard form $ax^2 + bx + c = 0$.

1. **State the problem:** We are given three points $(x, y)$ with $x$ values $-34$, $-17$, and $0$, and corresponding $y$ values $t$, $t + 29$, and $t + 58$, where $t$ is a constan

1. **State the problem:** We are given the system of equations: $$x - 3y = -9$$

1. The problem asks which numbers are greater than $$-\frac{185}{100}$$. 2. First, convert $$-\frac{185}{100}$$ to decimal: $$-\frac{185}{100} = -1.85$$.

1. **State the problem:** We are given the quadratic expression $p = n^2 + 21$ and need to find the value of $p$ for specific values of $n$. 2. **Formula:** The formula is $p = n^2

1. **State the problem:** Calculate the product of $\frac{7}{9}$ and $-\frac{9}{8}$. 2. **Formula used:** When multiplying fractions, multiply the numerators together and the denom

1. **Problem Statement:** Solve the determinant equation for part (a): $$\begin{vmatrix} 1 & 4 & 4 \\ 1 & -2 & 1 \\ 1 & 2x & x^2 \end{vmatrix} = 0$$

1. **State the problem:** Solve the system of linear equations: $$\begin{cases}-4r - 6s - 5t = 1 \\ -r + s - 2t = 2 \\ r - 3s + t = -5 \end{cases}$$

1. مسئله: بررسی درست یا نادرست بودن عبارت الف) معادله $x^4 - 5x^2 + 6 = 0$ دارای دو ریشه حقیقی متمایز است. 2. برای حل معادله $x^4 - 5x^2 + 6 = 0$، از جایگزینی $y = x^2$ استفاده می‌

1. مسئله: در محور اعداد نقطه A بین 2 و 3 قرار دارد. عددی که A نشان می‌دهد چیست؟ 2. برای تعیین مقدار A، باید موقعیت دقیق آن را روی محور اعداد بررسی کنیم. از نمودار مشخص است که A نزد

1. **Simplify using product law** (i) Simplify $\left(\frac{5}{4}\right)^2 \times \left(\frac{3}{7}\right)^4$

1. Let's start by stating the problem: Understanding ratio and proportion as explained in the 'The Leading' maths book for class eight. 2. A ratio is a comparison of two quantities