🧮 algebra

1. **State the problem:** Simplify the expression $$\left(\frac{3}{2}a + b\right)^3 - \frac{3}{2}a \left[\left(\frac{3}{2}a + b\right)^2 + (a - \frac{1}{4}b)(a + \frac{1}{4}b) - (a

1. **State the problem:** Find the mean of the two surds $\frac{1}{2\sqrt{2} - 1}$ and $\frac{1}{2\sqrt{2} + 1}$.\n\n2. **Recall the formula for the mean:** The mean of two numbers

1. The problem asks to find the range of values for $S$ given that $C$ ranges from 30 to 100, using the previous answer. 2. Assuming the previous answer established a relationship

1. **State the problem:** Rearrange the formula $$C=\frac{1000}{S+10}$$ to express $S$ in terms of $C$. 2. **Start with the given equation:**

1. Das Problem lautet: Berechne die Nullstellen der Funktion $$f(t) = 7t^2 - 22t + 3t^3 - 8t - 1 = \frac{1}{3}$$. 2. Zuerst bringen wir alle Terme auf eine Seite, um die Gleichung

1. **Problem:** Given $z = -3 + 4i$ and $w = 5 + 2i$, express $w$ in the form $\alpha + bi$ where $\alpha$ and $b$ are real numbers. **Step 1:** Identify the real and imaginary par

1. **Problem Statement:** Solve the first equation from SET 1: $5(x - 1) = 35$. 2. **Formula and Rules:** To solve linear equations, distribute multiplication over addition/subtrac

1. Problem: Solve for $x$ in the equation $x + 3y = 11$. 2. Formula: To solve for $x$, isolate $x$ on one side of the equation.

1. The problem asks for the slope and length of a line segment, but no specific points or line equation are given. 2. To find the slope between two points $(x_1, y_1)$ and $(x_2, y

1. **Problem Statement:** Sketch the graph of the quadratic function $f(x) = -2x^2 + 5x + 2$ and find its vertex, axis of symmetry, and intercepts.

1. **State the problem:** We are given several linear expressions and asked to analyze or simplify the first expression: $$\frac{2x - 12}{8}$$. 2. **Formula and rules:** To simplif

1. **Problem statement:** Find the intersection points of the parabolas in part (a): $$y = x^2 + 4x - 1$$

1. **State the problem:** Solve the equation $$\frac{2x + 1}{3} = -3$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side. Since $2x + 1$ is di

1. **State the problem:** Solve the equation $$\frac{x+4}{3} = -6$$ for $x$. 2. **Formula and rules:** To solve for $x$, multiply both sides of the equation by 3 to cancel the deno

1. Énoncé du problème : Calculer le déterminant de la matrice $$F=\begin{pmatrix}3m+1 & m-1 & m^2+1 \\ -5-m & 2m-2 & 2m \\ 2+2m & -m^2+2m & m+1\end{pmatrix}$$ en le factorisant le

1. **State the problem:** Simplify the expression $$\frac{x^2 - 9}{x^2 - 7x + 12} \div \frac{x}{5x - 20}$$ as a single fraction in fully factorised form. 2. **Rewrite division as m

1. **Énoncé du problème** : Calculer le déterminant de la matrice $F$ donnée par $$F = \begin{pmatrix} 3m + 1 & m - 1 & m^2 + 1 \\ -5 - m & 2m - 2 & 2m \\ 2 + 2m & -m^2 + 1 & m + 1

1. Il problema chiede di determinare per quali valori di $k$ l'equazione $x^2 - 3x + k = 1$ soddisfa alcune condizioni. 2. Riscriviamo l'equazione in forma standard:

1. **State the problem:** Solve for $x$ in the equation $$\sqrt{\frac{(x+2)!}{x!}} = \sqrt{3!} \times 7.$$\n\n2. **Recall factorial properties:** The factorial function $n!$ is the

1. **State the problem:** Evaluate the expression $5(-3)+(6-1)^2$. 2. **Apply order of operations:** First, calculate inside the parentheses and exponents.

1. **State the problem:** Evaluate the expression $5^2 - 4 \cdot 3 + \frac{28}{4}$. 2. **Recall order of operations:** Use PEMDAS (Parentheses, Exponents, Multiplication and Divisi