🧮 algebra

1. **State the problem:** We want to complete the proof for the formula

1. **State the problem:** Jennifer thinks of a number $h$. She triples it and then subtracts 11 to get an answer that is less than 34. 2. **Write the inequality:** Tripling $h$ mea

1. **State the problem:** We are given the equation $x(r - 7) + 3 = 17x + 25$ where $r$ is a positive integer. We want to find the values of $r$ for which the equation has exactly

1. **State the problem:** We are given the equation $$h = \frac{9(v - 273.15)}{5} + 32$$ which converts temperature from kelvins ($v$) to degrees Fahrenheit ($h$). We need to find

1. The problem is to expand the expression $ (3x - 2y)^2 $. 2. We use the formula for the square of a binomial: $$ (a - b)^2 = a^2 - 2ab + b^2 $$ where $a = 3x$ and $b = 2y$.

1. **State the problem:** Solve the inequality $x^2 - 5x + 6 < 0$. 2. **Recall the formula:** For quadratic inequalities, first find the roots of the quadratic equation $x^2 - 5x +

1. **State the problem:** We want to express $\frac{1}{20\sqrt{20}}$ in the form $\frac{\sqrt{5}}{n}$, where $n$ is a positive integer. 2. **Rationalise the denominator:** Start wi

1. Дано равенство: $x^2 - 5x + 6 = (x - 2)(x - 3)$. 2. Это равенство показывает, что многочлен второй степени $x^2 - 5x + 6$ можно разложить на произведение двух линейных множителе

1. **State the problem:** Solve the equation $$\frac{5x^2}{2} + 47 = 137$$ for $x$. 2. **Isolate the term with $x^2$:** Subtract 47 from both sides:

1. **State the problem:** Solve the inequality $x^2 - 5x + 6 < 0$. 2. **Recall the formula:** This is a quadratic inequality. We first find the roots of the quadratic equation $x^2

1. **State the problem:** Simplify the expression $10 - (4 + 2) \times 2$. 2. **Recall the order of operations:** Parentheses first, then multiplication, and finally subtraction.

1. **State the problem:** Solve the equation $2-\frac{1}{2}n=3n+16$ for $n$. 2. **Write down the equation:**

1. **State the problem:** Solve the equation $9e+4 = -5e+14 + 13e$ for $e$. 2. **Rewrite the equation:** The equation is $9e + 4 = -5e + 14 + 13e$.

1. **بيان المسألة:** لدينا الدالة $$f(x) = x \sqrt{x^2 - 1}$$ ونريد دراسة سلوكها على المجال $$I = [1, +\infty[\,$$ ومقارنتها مع الخط المستقيم $$y = x$$. 2. **فهم الدالة:** الدالة ت

1. **بيان المسألة:** حل المعادلة في مجموعة الأعداد الحقيقية IR:

1. **بيان المسألة:** لدينا الدالة $$f(x) = x \sqrt{x^2 - 1}$$ ونريد دراسة خصائصها ضمن المجال $$I = ]-\infty, +1]$$. 2. **تحديد المجال:** الجذر التربيعي يتطلب أن يكون التعبير داخل ا

1. Let's start by stating the problem: We want to understand why a given equation represents a hyperbola. 2. The general form of a hyperbola centered at the origin is given by the

1. **Problem Statement:** Given the conic section equation $$9x^2 - 4y^2 + 18x - 8y + 6 = 0,$$ we need to:

1. Das Problem lautet: Finde die kubische Funktion $f(x) = ax^3 + bx^2 + cx + d$, die durch den Ursprung geht, bei $x=4$ ein Minimum hat mit Funktionswert $-80/3$ und bei $x=-2$ ei

1. **State the problem:** Solve the quadratic equation $x^2 + 6x - 5 = 0$ for $x$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c

1. The problem states that $x$, 17, $3x - y^2 - 2$, and $3x + y^2 - 30$ are four consecutive terms of an increasing arithmetic sequence. 2. In an arithmetic sequence, the differenc