🧮 algebra

1. **State the problem:** Complete the square for the quadratic equation $$x^2 - 8x = 10$$ given constants $$A=13$$ and $$C=-20$$. 2. **Recall the formula for completing the square

1. **State the problem:** We want to find the price per phone $x$ that maximizes the revenue function $$r(x) = x \cdot (1200 - 2x)$$ where $x$ is the price per phone in dollars. 2.

1. **Problem Statement:** Given a quadratic function, factor it, find the x-intercepts, axis of symmetry, and vertex. 2. **General Form and Formula:** A quadratic function is gener

1. **Problem 1: Pattern of squares** We are given a pattern where $S$ represents the number of small squares as a function of the step number $n$. The chart shows some values of $S

1. **Stating the problem:** We need to factorize a given algebraic expression. Since the user did not specify the expression, let's consider a common example: $$x^2 - 5x + 6$$. 2.

1. **Problem Statement:** Find the x-intercepts, axis of symmetry, and vertex for each quadratic equation given in factored form. 2. **Formula and Rules:**

1. The problem is to analyze and simplify the expression: $$a\alpha^{3} - (c - a)^{2}x^{2} - 2\alpha x + 1$$. 2. We start by understanding the terms and their powers. The expressio

1. The problem is to simplify the expression $2$. 2. Since $2$ is already a simplified number, no further operations are needed.

1. **State the problem:** Given that $x$ varies inversely as $y$, and $x=4$ when $y=8$, find $x$ when $y=1$. 2. **Formula and rule:** For inverse variation, the relationship is giv

1. **State the problem:** Solve for $x$ in the equation $3x - 7 = 2x + 5$. 2. **Write down the equation:**

1. **State the problem:** If $x$ varies inversely as $y$, and $x=4$ when $y=8$, find $x$ when $y=1$.

1. **Problem K:** Simplify and solve $$5x^2 \cdot (3x^2 + 1)^4 \cdot 6x + (3x^2 + 1)^5 \cdot 2x$$. 2. **Step 1:** Write the expression clearly:

1. **State the problem:** We are given the function $f(x) = x^2 + 25$ and need to find the value of $x$ such that $f(x) = 29$. 2. **Write the equation:** Set $f(x)$ equal to 29:

1. **State the problem:** We are given two simultaneous linear equations: $$\text{(1) } 5e + f = 16$$

1. **State the problem:** We are given two linear equations: $$y = 3x - 4$$

1. **State the problem:** We are given two linear equations: $$y = \frac{2}{3}x - 1$$

1. The problem is to create the graph of a function that can be copied down easily. 2. Since no specific function was given, let's consider a simple example: the quadratic function

1. The problem asks to create a graph for the function y = 6. 2. This is a constant function where the value of y is always 6 regardless of x.

1. **Planteamiento del problema:** Simplificar completamente la expresión $$\left[\begin{array}{cc}\frac{2}{x-1} - \frac{3}{x+2} \\\frac{1}{x-1} + \frac{1}{x+2}\end{array}\right] \

1. **State the problem:** Find the points of intersection between the circle $$x^2 + y^2 = 25$$ and the line $$y = 2x - 12$$. 2. **Formula and rules:** To find intersection points,

1. **State the problem:** Find the points of intersection between the circle defined by $$x^2 + y^2 = 25$$ and the line $$y = 2x - 12$$. 2. **Formula and rules:** To find intersect