🧮 algebra

1. **Problem Statement:** We want to prove the formula for the sum of the first $n$ squares:

1. The problem asks to write the number $4.58 \times 10^{-3}$ as an ordinary decimal number. 2. The scientific notation $a \times 10^{n}$ means move the decimal point in $a$ by $n$

1. The problem asks to write the number 6 050 000 in standard form. 2. Standard form (also called scientific notation) expresses a number as $a \times 10^n$ where $1 \leq a < 10$ a

1. **State the problem:** Solve the inequality $$2x + 5 \geq 11$$ and select the correct solution on the number line. 2. **Write the inequality:** $$2x + 5 \geq 11$$

1. We are asked to simplify the following quadratic expressions: (4) \quad 3x^2 + 7xy + 2y^2 + 11x + 7y + 6

1. **Problem 1:** Factorise $$2r^2 + 7rxy + 3y^2 + 9x + 7y + 4$$. 2. Group terms: $$2r^2 + 7rxy + 3y^2 + 9x + 7y + 4 = 2r^2 + (7ry) x + (3y^2 + 7y + 4) + 9x - 7y$$.

1. **State the problem:** Solve the system of linear equations: $$5.109x + 7y = 0$$

1. Let's clarify the problem: you want to know what to put in the squares, which usually means filling in missing values or expressions in a math problem involving squares or squar

1. **State the problem:** We want to approximate the value of the function $$f(x) = x^3 - 3x^2 + 4x + 0.21$$ at $$x = 2.15$$ using nested arithmetic and 3-digit rounding. 2. **Rewr

1. **State the problem:** Solve the system of linear equations: $$-4.10x + 7y = 0$$

1. The problem is to solve the system of simultaneous equations. 2. A system of simultaneous equations consists of two or more equations with multiple variables that are solved tog

1. The problem is unclear as "r er 5" does not form a recognizable mathematical expression or question. 2. Please provide a clear algebraic expression or a specific question to sol

1. **Arithmetic Progressions (AP) Problem:** State the problem: Find the 10th term and the sum of the first 10 terms of an AP where the first term $a_1=3$ and common difference $d=

1. **State the problem:** We need to calculate the overall rating of an employee based on weighted scores in three categories: Job knowledge (50%), Teamwork and loyalty (35%), and

1. The problem is to analyze the function $y=\log_{\frac{1}{2}} x$. 2. The logarithm function $y=\log_a x$ is defined for $x>0$ and base $a>0$, $a \neq 1$.

1. **State the problem:** We need to solve the system of linear equations: $$\alpha_1 + \alpha_2 + \alpha_3 = 7$$

1. **Problem Statement:** Solve the three sets of simultaneous equations with three variables $x$, $y$, and $z$. ---

1. We are given three systems of simultaneous equations to solve for variables $x$, $y$, and $z$. 2. Let's start with system (1):

1. **State the problem:** Solve the equation $$\sqrt{3}(x - 2\sqrt{3}) = x + 2\sqrt{3}$$ for $x$. 2. **Recall the distributive property:** Multiply $\sqrt{3}$ by each term inside t

1. **State the problem:** We are given two functions $f(x) = 2x + 3$ and $g(x) = \frac{x^2 - g}{x - 3}$, and we need to find $f(g(x))$ at $x = 3$. 2. **Clarify the function $g(x)$:

1. **Stating the problem:** We are given a function defined as $f\left(\frac{1}{x}\right) = \frac{x^4 + x^2 - 1}{x^2 - x}$ and asked to find the expression for $f(x)$. 2. **Underst