🧮 algebra

1. Let's start with problem (a): Find the set $A_1 = \{x \in \mathbb{Z} : 2x^2 - 3 = x\}$. 2. Rearrange the equation:

1. Planteamos el problema: Encontrar los valores de $A$ y $B$ para la función por partes $$Z(t) = \begin{cases} t^2 + At + 5, & 1 \leq t \leq 4 \\ B - t, & 4 < t \leq 10 \end{cases

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} -x^2 + 10 & \text{if } -4 \leq x < 4 \\ 6 - 3x & \text{if } x \geq 4 \end{cases}$$

1. **State the problem:** Solve the equation $$\frac{3}{x+1} - \frac{2}{x^2-1} = \frac{1}{x^2 - x}$$ for $x$. 2. **Identify the denominators and factor where possible:**

1. The problem states that the $r$th term of the first sequence is $2r^2 - 1$ and the $r$th term of the second sequence is $40 - r^2$. We need to show there is only one number comm

1. **State the problem:** We have two rectangles A and B. Rectangle A has dimensions $4x$ cm and $2.5$ cm.

1. **State the problem:** We have a trapezium ABCD with sides AD = $x$ cm, BC = $x$ cm, AB = $2x$ cm, and DC = $2x + 4$ cm. The perimeter is 38 cm. We need to find the length of AD

1. **Énoncé du problème :** Nous avons la fonction $g$ définie par $g(x) = x^3 - x^2 + 3x + 1$.

1. Staðfesta verkefnið: Reikna og skila sem fullstyttu broti fyrir a) $\frac{3a^3 \cdot b^2}{2b} \cdot \frac{4b}{6a^2}$ og b) $\frac{y^2}{x \cdot z} \cdot \frac{y}{z^2 \cdot x^3}$.

1. Staðfesta verkefnið: Þáttaðu liðastærðir og fullstyttu brotin. 2. Við byrjum á liðunum í 1. lið:

1. **State the problem:** We are given the function $f(t) = t$ defined on the interval $0 < t < \pi$. We want to analyze or solve for properties related to this function. 2. **Form

1. **Problem:** Find the function expressions for the given intervals. 2. **Step 1:** For $f(t) = t$, $0 < t < \pi$.

1. **Problem statement:** We are given three functions: - $f(x) = x^4 + x + 2$

1. **State the problem:** Find factors of 8 and 20 such that when multiplied and then added, the sum is 14. 2. **Identify factors:**

1. **Stating the problem:** We need to find the number of ordered positive integer pairs $(a,b)$ such that $$a^2 - b^2 = 2025.$$ 2. **Formula and important rules:** Recall the diff

1. **Problem statement:** We need to find the number of ordered positive integer pairs $(a,b)$ such that $$a^2 - b^2 = 2026.$$ 2. **Formula and approach:** Recall the difference of

1. **Problem statement:** Simplify and evaluate numerical expressions involving positive and negative integer indices. 2. **Formula and rules:** For any nonzero number $a$ and inte

1. **State the problem:** Solve the equation $$7 = \frac{b}{3} - 5b + 2$$ for $b$. 2. **Rewrite the equation:** Move all terms to one side to isolate $b$ terms.

1. **Problem statement:** Evaluate and simplify numerical expressions raised to fractional indices. 2. **Formula and rules:** A fractional exponent $a^{\frac{m}{n}}$ means the $n$t

1. **State the problem:** Given $x = \sqrt{78}$ and $y = \sqrt{77}$, find the value of $x + y - \sqrt{x^2 + y^2}$.\n\n2. **Recall the values:**\n$x = \sqrt{78}$\n$y = \sqrt{77}$\n\

1. The problem is to find the distance between two points or objects. 2. The most common formula to find distance when you know speed and time is: