🧮 algebra

1. المشكلة: لدينا دالة $x_0 = \exp(-x^2)$ ونريد فهم العلاقة بين $x_0$ و $\exp(-(x-1)^2)$ مع بعض المعادلات المتعلقة بالقيم $x$. 2. نبدأ بفهم المعادلة المعطاة:

1. **State the problem:** We have a table with two days, Monday and Tuesday, showing numbers of flies and mosquitoes. We know Monday has 15 flies and Tuesday has 14 mosquitoes. We

1. مسئله: نمودار تابع $y = x^{3}$ را از انتقال افقی یا عمودی به نمودار $y = 3x^{3}$ تبدیل کنید. 2. فرمول و توضیح: تابع $y = 3x^{3}$ را می‌توان به صورت $y = 3 imes (x^{3})$ نوشت. ا

1. **State the problem:** Factor the quadratic expression $x^2 - 7x + 10$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multip

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** Simplify the expression $$\sqrt{24} + \sqrt{150} + \sqrt{54}$$ without using a calculator. 2. **Recall the rule:** The square root of a product can be wri

1. **State the problem:** Simplify the expression $$(\sqrt{2} - \sqrt{3})^2$$. 2. **Recall the formula:** The square of a binomial $(a - b)^2$ is given by $$a^2 - 2ab + b^2$$.

1. **Problem:** Determine whether the function $f(x)$ is one-to-one (injective). 2. **Definition:** A function is one-to-one if each $y$-value corresponds to exactly one $x$-value.

1. **Problem statement:** Given the function $f : \mathbb{R} \to [-1, \infty)$ defined by $f(x) = 4x^2 - 1$, find the domain and range of $f$ and sketch its graph. 2. **Domain:** T

1. **Problem (i) Part 1:** Find the range of values of $x$ for which $$\frac{2x+1}{x-2} < 1.$$ **Step 1:** Start with the inequality: $$\frac{2x+1}{x-2} < 1.$$

1. **Problem 1:** Calculate $$\sqrt{(0.42-0.424)^2+(0.48-0.424)^2+(0.42-0.424)^2+(0.4-0.424)^2+(0.4-0.424)^2}$$ and then divide the result by 2. 2. **Step 1:** Calculate each squar

1. Problem 13: Number of Ducks and Pigs We are given that the total number of ducks and pigs is 35 and the total number of legs is 98. Each duck has 2 legs and each pig has 4 legs.

1. **Stating the problem:** We have a polynomial $f(x)$ such that

1. **Problem:** Given the quadratic equation $$2x^2 - 5x + 2 = 0$$ with roots $$p$$ and $$q$$, show that $$2p + p^2 q + 2q - (pq)^2 = \frac{13}{2}$$. 2. **Formula and rules:**

1. The problem is to rewrite the expression with the left side as $16|9(9^k+1-8k-17)$ and the rest divided by 16. 2. First, understand the notation: $16|9(9^k+1-8k-17)$ means $16 \

1. **Stating the problem:** Prove by induction that $16 \mid 9^{n+1} - 8n - 9$ for all integers $n \geq 0$. 2. **Base case:** For $n=0$, check if $16 \mid 9^{0+1} - 8\cdot0 - 9 = 9

1. Énoncé du problème : Évaluer l'expression $2ac$ où $a$ et $c$ sont des nombres rationnels donnés. 2. Formule utilisée : Pour évaluer $2ac$, on multiplie simplement les nombres $

1. Planteamos el problema: Dadas las funciones $f(x) = 4x^2 + 6x - 11$ y $(f \circ g)(x) = 36x^2 - 30x - 7$, debemos encontrar el valor de $g\left(\frac{2}{3}\right)$. 2. Recordemo

1. Planteamos el problema: Resolver la ecuación $$\log_2 x - \log_x 8 = 2$$. 2. Recordemos que el cambio de base para logaritmos es $$\log_a b = \frac{\log_c b}{\log_c a}$$ para cu

1. The problem asks to find the value of $$(3^2 - 2^4)^2$$. 2. First, calculate the powers inside the parentheses: