🧮 algebra

1. **State the problem:** Solve the quadratic equation $$2x^2 - 19 = 0$$ by completing the square. 2. **Rewrite the equation:** Move the constant term to the right side:

1. Masalah ini meminta kita menentukan nilai $x$ yang memaksimalkan luas taman. 2. Misalkan fungsi $y$ dinyatakan dalam $x$. Biasanya, jika taman berbentuk persegi panjang dengan p

1. Planteamos el problema: hallar el área menor encerrada por las curvas dadas: $$x^2 - y = 0 \implies y = x^2$$

1. **Problem:** Solve the system of linear equations: $$\begin{cases} x - y - 2z = 0 \\ -2x + 6y = 1 \\ -x + 2y + z = 1 \end{cases}$$

1. **Problem statement:** Find the domain (conditions for which the function is defined) of each rational expression. 2. **Key rule:** A rational expression is defined when its den

1. The problem is to solve a word problem involving algebraic expressions or equations. 2. First, identify the unknowns and assign variables to them.

1. State the problem: Solve the linear equation $2x + 3y = 6$ for $y$. 2. Formula: A linear equation in two variables $x$ and $y$ is generally written as $ax + by = c$ where $a$, $

1. **State the problem:** We are given the equation $$\frac{K + K}{K} = 6$$ and need to find the value of $K$. 2. **Rewrite the numerator:** Note that $K + K = 2K$.

1. **State the problem:** Solve the quadratic equation $x^2 + 6x + 6 = 0$ using the square (completing the square) method. 2. **Formula and rules:** The square method involves rewr

1. **State the problem:** We are given the function $$g(x) = 1 - 2(x+2)^2 - \ln(x+2)$$ and we want to understand or analyze it. 2. **Rewrite the function clearly:**

1. **State the problem:** Solve the equation $$\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2x}$$ for $x$. 2. **Formula and rules:** To solve rational equations, first find a common de

1. **Problem 4:** Find values of $k$ such that the equation $$p(x+1) + q(x-r) = \frac{k}{2}x$$ has two equal roots. 2. **Rewrite the equation:**

1. **Problem statement:** Find the value of $m$ such that the system of equations

1. **Problem:** Find the number of intersection points between the parabola $y=3x^2$ and the line $y=3x+5$. 2. **Set the equations equal to find intersection points:**

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle that to isolate $x$, we perform inverse operations to both sides of the equati

1. Vamos calcular o item a) do exercício: $\left(\frac{3}{7}\right)^2 \cdot \frac{16}{9}$. 2. Primeiro, elevamos $\frac{3}{7}$ ao quadrado:

1. **State the problem:** Simplify and evaluate the expression $$\left(3^3 - \frac{1}{5^{-2}}\right)^{-4}$$. 2. **Recall the rules:**

1. **State the problem:** We need to find the sum $$\sum_{i=1}^{20} \left( x^2 \sqrt{\frac{9x^4}{x^8}} \sqrt[i]{4} + \sqrt{\frac{125}{5}}^{2 \log_5 5} + \log_3 243 \right)$$. 2. **

1. نبدأ ببيان المشكلة: لدينا معادلة تربيعية جذورها أحدها هو $2+i$. 2. بما أن المعاملات حقيقية، فإن الجذر الآخر هو المرافق المركب لـ $2+i$، أي $2 - i$.

1. **State the problem:** Solve the inequality $$x^2 + 7x + 12 \geq 0$$. 2. **Formula and rules:** To solve quadratic inequalities, first find the roots of the quadratic equation $

1. **Problem:** Solve the inequality $x^2 - 3x + 2 > 0$. 2. **Formula and rules:** For quadratic inequalities of the form $ax^2 + bx + c > 0$, first find the roots by solving $ax^2