🧮 algebra

1. The problem: Solve the quadratic equation $2x^2 - x - 1 = 0$ using the quadratic formula. 2. The quadratic formula is:

1. Simplify \(\sqrt{98}\): \(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}\).

1. **Énoncé du problème** : On considère la suite $(s_n)$ définie par $$s_n = 1 + \frac{1}{3} + \frac{1}{3^2} + \cdots + \frac{1}{3^n}.$$

1. Se plantea resolver cada ecuación logarítmica dada paso a paso. **a)** $$\log_2 2 + \log(11 - x^3) = \frac{2}{\log(5 - x)}$$

1. The problem states the quadratic function as $A(x) = ax^2 + bx + c$. 2. This is a general form of a quadratic equation where $a$, $b$, and $c$ are constants and $x$ is the varia

1. We are given the quadratic function $F(x) = x^2 - 6x + 5$. 2. We want to express it in the form $F(x) = (x + a)^2 + b$ where $a$ and $b$ are constants.

1. The problem is to factor the quadratic polynomial $2x^2 - x - 1$. 2. We look for two numbers that multiply to $2 \times (-1) = -2$ and add to $-1$ (the coefficient of $x$).

1. **مشكلة:** تقاسم ثلاثة إخوة أرباح تجاراتهم \n- الأول أخذ \( \frac{1}{5} \) من الأرباح \n- الثاني أخذ \( \frac{3}{10} \) من الأرباح \n- الثالث أخذ \( \frac{1}{4} \) مما أخذ الأول

1. **Definición del logaritmo:** El logaritmo de un número $x$ en base $b$ se define como el exponente al que hay que elevar $b$ para obtener $x$. Esto se expresa como: $$\log_b x

1. State the problem: Express $$\frac{(\log m)^p \sqrt{3 - n}}{m n^2}$$

1. **State the problem:** Express $$\frac{\log m^p \sqrt{3-n}}{mn^2}$$ in terms of $$\log m$$, $$\log n$$, and $$\log p$$. 2. **Rewrite the numerator:**

1. Stating the problem: We are given the function $F(x) = x^2 - 6x + 5$ and need to analyze it. 2. Identify the type of function: This is a quadratic function in standard form $F(x

1. **State the problem:** Find the domain and range of the function $$y=\frac{\sqrt{x^2 - 5x + 6}}{\sqrt{x^2 - x + 6}}.$$\n\n2. **Domain analysis:** The domain consists of all $x$

1. The problem is to analyze the quadratic function $F(x) = x^2 - 6x + 5$. 2. First, find the vertex to understand the function's shape. The vertex formula for $x$ is given by $x =

1. **State the problem:** Express $$\frac{(\log m)^p \sqrt{3 - n}}{m n^2}$$ in terms of \(\log m\), \(\log n\), and \(\log p\). 2. **Analyze each part:**

1. Resuelve la ecuación $\log 2 + \log(11 - x^2) = 2 \log(5 - x)$ Usamos propiedades de los logaritmos: $\log a + \log b = \log(ab)$ y $n \log a = \log(a^n)$.

1. Énoncé du problème : Soit la suite $(s_n)$ définie par

1. We are given the quadratic function $F(x) = x^2 - 6x + 5$. 2. To analyze this function, let's find its vertex by completing the square.

1. Resolver la ecuación \(\log 2 + \log(11 - x^2) = 2 \log(5 - x)\) Usamos la propiedad de logaritmos \(\log a + \log b = \log(ab)\) y \(2 \log c = \log c^2\):

1. The problem is to solve the quadratic equation $$3x^2 - 4x + 9 = 0$$. 2. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $$a=3$$, $$b=-4$$, and $$c=

1. Problem 14(a): Simplify $\sqrt{32} + \sqrt{98}$. - $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$.