🧮 algebra

1. Problem (2a)(i): Simplify the expression $$\frac{(1 + x)^{\frac{1}{2}} - \frac{1}{2} x (1 + x)^{-\frac{1}{2}}}{(1 + x)^{\frac{3}{2}}}$$

1. The problem is to express the numbers 6 and 8 as fractions. 2. Any integer can be written as a fraction by placing it over 1. This is because dividing by 1 does not change the v

1. Stating the problem: Factorize the quadratic expression $x^2 + 8x + 7$. 2. Formula and rules: To factor a quadratic expression of the form $ax^2 + bx + c$, we look for two numbe

# Linear Equations in One Variable ## Introduction

1. **Problem statement:** Simplify the expression $3\sqrt{20} - 2\sqrt{10}$. 2. **Recall the rule:** To simplify expressions with square roots, first express each radical in terms

1. The problem is to find the cube root of $-1$. 2. The cube root of a number $x$ is a number $y$ such that $y^3 = x$.

1. **State the problem:** A group of people bought a 300 ha farm and shared it equally. The number of hectares per person is 5 less than the number of people. We need to find the n

1. সমস্যাটি হলো: একটি সরলরেখার সমীকরণ নির্ণয় করা যা বিন্দু (2, 3) দিয়ে যায় এবং ঢাল $m=4$। 2. সরলরেখার সমীকরণের সাধারণ ফর্ম হলো: $$y - y_1 = m(x - x_1)$$ যেখানে $(x_1, y_1)$ হলো

# Linear Equations in One Variable ## Understanding Linear Equations

1. **Stating the problem:** We will learn about linear equations in one variable, which are equations that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants

1. Let's start by understanding the problem you are referring to. Since you mentioned "the last one," please provide the exact problem statement or the equation you want to solve.

1. **State the problem:** We have a rectangle with original length $8$ cm and width $5$ cm. Both length and width are reduced by the same amount $x$ cm, and the area of the new rec

1. **Problem 8:** Determine how many solutions the equation $f(x) = g(x)$ has given the graphs of $f$ and $g$. 2. The solutions to $f(x) = g(x)$ are the $x$-values where the graphs

1. **Problem 4:** Find the equation of the line passing through (0, -3) parallel to the line given by options A-D. 2. The given lines have the form $2x - 3y = c$. To find the slope

1. **State the problem:** We are given two lines, $L_1$ with equation $x + 2y = 4$ and $L_2$ passing through points $(-1, -7)$ and $(7, 9)$. We need to determine if $L_1$ and $L_2$

1. **State the problem:** Solve the equation $$\sqrt{10} - 3\sqrt{5} = 3\sqrt{5} - 2$$ for the unknowns or verify the equality. 2. **Rewrite the equation:** The equation is $$\sqrt

1. **Problem Statement:** We are given the ratio of the difference between Compound Interest (C.I) and Simple Interest (S.I) for 3 years and 2 years as 10:3. We need to find the ra

1. The problem is to express a given condition or example in set-builder notation. 2. Set-builder notation describes a set by stating the properties that its members must satisfy.

1. **State the problem:** We need to find the term number $n$ in an arithmetic progression (A.P.) where the first term $a_1=3$, the common difference $d=7$, and the $n$th term $a_n

1. **Stating the problem:** We are given the function $f(x) = x^2 \cdot (x - 5)^4$ and asked to analyze its local extrema. 2. **Formula and rules:** To find local extrema, we first

1. **Problem Statement:** Find the general term, the 10th term, and the first three terms of an arithmetic progression (A.P.) where the first term $a_1=4$ and the common difference