🧮 algebra

1. **State the problem:** We are given the sequence 2, 6, 12, 20, 30 and asked to find the next number. 2. **Identify the pattern:** Let's look at the differences between consecuti

1. **State the problem:** We are given the sequence 2, 6, 12, 20, 30 and asked to analyze it. 2. **Identify the pattern:** Notice the terms: 2, 6, 12, 20, 30.

1. **State the problem:** We are given the equation $$K = \sqrt{\frac{3 + 5m^2}{2m}}$$ and asked to solve for $m$. 2. **Square both sides** to eliminate the square root:

1. The problem is to identify the pattern in the sequence: 2, 5, 11, 23, 47. 2. We observe the differences between consecutive terms: 5 - 2 = 3, 11 - 5 = 6, 23 - 11 = 12, 47 - 23 =

1. The problem is to find the next number in the sequence: 6, 12, 21, 33. 2. First, observe the differences between consecutive terms:

1. **State the problem:** We need to find how many odd numbers are there between 16 and 80. 2. **Understand the problem:** Odd numbers are integers that are not divisible by 2. We

1. **Problem Statement:** An element with an initial mass of 430 grams decays by 27.4% per minute. We want to find how much of the element remains after 19 minutes, rounded to the

1. **State the problem:** Solve the equation $$2x^2 + xy - y^2 = 2$$ for $y$ in terms of $x$. 2. **Identify the type of equation:** This is a quadratic equation in $y$ with terms i

1. **Problem:** Solve the proportion $$\frac{2x + 3}{2} = \frac{6x - 3}{5}$$ for $x$. 2. **Formula and rules:** To solve proportions, cross-multiply:

1. **State the problem:** Find the inverse function of $g(x) = x^3$. 2. **Recall the definition of inverse function:** The inverse function $g^{-1}(x)$ satisfies $g(g^{-1}(x)) = x$

1. **Énoncé du problème** : Résoudre dans $\mathbb{R}$ l'équation $\frac{3x+1}{x} = \frac{2 - x}{2}$. 2. **Formule et règles importantes** : Pour résoudre une équation de fractions

1. **State the problem:** Find the inverse function of $h(x) = x^2 - 2x + 1$. 2. **Rewrite the function:** Notice that $h(x) = x^2 - 2x + 1$ can be rewritten by completing the squa

1. **Énoncé du problème :** Résoudre dans $\mathbb{R}$ l'équation $$\frac{3x+1}{x} = \frac{2-x}{2}.$$ 2. **Formule et règles importantes :** Pour résoudre une équation de fractions

1. **State the problem:** Find the inverse function of $f(x) = 2x + 1$. 2. **Recall the formula and rules:** The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find

1. The problem is to solve the equation $$2^x = x^2$$ for the value(s) of $x$. 2. This is a transcendental equation where an exponential function equals a polynomial function. Such

1. **State the problem:** Compare the two fractions $\frac{111}{1111}$ and $\frac{1111}{11111}$ to determine which is greater. 2. **Formula and approach:** To compare two fractions

1. **Problem 1:** Calculate $a + b + c + d + e + f$ if $abcdef = 6(defabc)$. 2. We are given the equation:

1. The problem asks to find which fraction is equivalent to the mixed number $2 \frac{3}{4}$. 2. First, convert the mixed number to an improper fraction.

1. نبدأ بحل التمرين الأول: بسط الأعداد التالية ثم نذكر أصغر مجموعة ينتمي إليها كل عدد. 2. العدد الأول: $$\frac{2}{\sqrt{3}+1}$$

1. **State the problem:** Find the value of $x$ that satisfies the equation $$\frac{2x-3}{5} = 7.$$\n\n2. **Formula and rules:** To solve for $x$ in a linear equation with a fracti

1. **Problem Statement:** We are given the function $f(x) = \sqrt[5]{x}$ and its transformations: - $f(x) = \sqrt[5]{x} + 6$