🧮 algebra

1. **State the problem:** We need to expand and simplify the expression $(-7x+2)(-3x-4)$. 2. **Formula used:** To multiply two binomials, use the distributive property (FOIL method

1. **State the problem:** Simplify the expression $$(9x+3)(-2x-2)$$. 2. **Formula used:** Use the distributive property (also known as FOIL for binomials):

1. **State the problem:** Find the zeros (roots) of the quadratic function given in vertex form: $$a(x-h)^2 + k = 0$$ 2. **Formula and rules:** The zeros of a quadratic are the val

1. Énonçons le problème : Trouver les coefficients du polynôme caractéristique d'une matrice donnée en utilisant la méthode des coefficients. 2. Le polynôme caractéristique $p(\lam

1. Énonçons le problème : On cherche à vérifier le calcul du polynôme caractéristique $\chi_A(\lambda)$ de la matrice symétrique donnée et à comprendre pourquoi le résultat ne corr

1. **Problem statement:** Stelle eine Gleichung auf und löse mit quadratischer Ergänzung: Das doppelte Quadrat einer Zahl ist um eins größer als die Differenz aus dem 24-fachen der

1. **State the problem:** A boat travels 25 km upstream in 5 hours and returns downstream in 0.5 hours. We need to find the speed of the boat in still water ($b$) and the speed of

1. **State the problem:** We are given the formula for the Ponderal Index: $$P = \frac{M}{h^3}$$ where $P$ is the Ponderal Index, $M$ is the mass (weight), and $h$ is the height.

1. **Stel het probleem vast:** We willen de richtingsgetallen (richtingsvector) van de rechte $f$ vinden, gegeven in parametervergelijking: $$f : \begin{cases} x = 1 + 2k \\ y = 10

1. **Problemstellung:** Wir sollen 1 Liter der neuen Sorte PLOP mit 50% Gesamtfruchtgehalt und 20% Maracujaanteil aus den drei Mischgetränken A, B und C mischen. 2. **Gegebene Fruc

1. **Problem statement:** We want to find if a mixture of components A, B, and C can produce a food with given percentages of protein (Eiweiß), carbohydrates (Kohlenhydrate), and f

1. **Stating the problem:** We are given that 1 kilometer equals approximately 0.621 mile.

1. **Problem statement:** A tablet screen measures 8 inches by 5 inches, and it has a border of uniform thickness $x$ around it. We need to find expressions and solve for $x$ based

1. The problem is to evaluate fractional exponents and match them with their numerical values. 2. Recall the rule for fractional exponents: $$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a}

1. **State the problem:** Solve the quadratic equation $$3x^2 - 5x + 2 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is

1. **State the problem:** Solve the quadratic equation $$x^2 + 2x - 3 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions ar

1. **State the problem:** We need to find the sum of the infinite series $$\sum_{n=1}^\infty \frac{e^n}{\pi^n}$$ or determine if it diverges. 2. **Identify the type of series:** Th

1. **State the problem:** We need to find the value of the constant $k$ such that the equation $$\frac{kx^2 + 14x - 20}{3x - 2} = 5x + 8 - \frac{4}{3x - 2}$$ holds true for all $x

1. **State the problem:** Solve the rational equation $$\frac{9}{2} = \frac{7x - 8}{x + 1}$$ and check for extraneous solutions. 2. **Formula and rules:** To solve rational equatio

1. **State the problem:** Solve the rational equation $$\frac{3}{x - 7} = \frac{x + 1}{3}$$ and check for extraneous solutions. 2. **Identify the domain restrictions:** The denomin

1. **Problemstellung:** Auf einem Schachbrett mit 64 Feldern verdoppelt sich die Anzahl der Reiskörner von Feld zu Feld, beginnend mit 1 Reiskorn auf dem 1. Feld.