🧮 algebra

1. **State the problem:** We are given the x-intercepts of a parabola at $(-2,0)$ and $(3,0)$ and the vertex at $(0.5,-12.5)$. We need to find the equation of the parabola. 2. **Fo

1. **State the problem:** Find the value of $x$ when $f(x)$ is 4 more than $g(x)$. 2. **Write the functions:**

1. **State the problem:** We want to analyze and understand the function $$y = \frac{1 + x^2}{x}$$. 2. **Rewrite the function:** We can split the fraction to simplify understanding

1. **Problem statement:** Calculate the profit percentage when an article costing 612 is sold at a marked price of 1275 with a 28% discount.

1. **State the problem:** We know that $x$ is inversely proportional to $y$. Given $y=27$ when $x=14$, find $x$ when $y=21$. 2. **Formula and rule:** If $x$ is inversely proportion

1. **Problem Statement:** Find the function and analyze the expression $y = (x + 1)^3 - 1$. 2. **Formula and Rules:** This is a cubic function shifted horizontally and vertically.

1. **Stating the problem:** Find the result of $$\frac{(3^3 p^4)^3}{3 p^2}$$.

1. The problem asks us to identify which number in the set $\{-49, \frac{7}{22}, 3\pi, -2.5, 0.1\dot{2}\}$ is irrational. 2. Recall that an irrational number is a number that canno

1. **Problem Statement:** Given the graph of $y = f(x)$ with vertices at $(-6, -2)$, $(-2, 4)$, and $(2, 2)$, we need to draw the graph of $y = -f(x)$. 2. **Understanding the Trans

1. **State the problem:** Calculate the value of the expression $4640 \times 0.083 + 0.0000020 \times 5585000 \sqrt{0.083}$.\n\n2. **Recall formulas and rules:**\n- Multiplication

1. The problem is to calculate the value of $r_1^*$ given by the expression $r_1^* = 4.640 \times 0.083 + 0.0000020 \times 5,585,000 \times \sqrt{0.083}$. 2. First, multiply $4.640

1. مسئله را بیان می‌کنیم: مجموع ریشه‌های معادله $$8 - x = \sqrt{x} - 3x - 9\sqrt{x} - x^9$$ را پیدا کنیم. 2. ابتدا معادله را بازنویسی می‌کنیم:

1. **Stating the problem:** Given the quadratic equation $$2y^2 - 5y - 3 = 0$$, find the roots \(\alpha\) and \(\beta\). 2. **Formula used:** For a quadratic equation $$ay^2 + by +

1. مسئله را بیان می‌کنیم: اگر $x=\alpha$ ریشه معادله $$\frac{4}{x-2} = \frac{\sqrt{x}}{\sqrt{x} + \sqrt{2}}$$ باشد، مقدار $$\sqrt{2\alpha - 4}$$ را بیابید. 2. ابتدا معادله را ساده

1. The problem states that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2y^2 - 5y - 3 = 0\). We need to find the value of \(\alpha + \beta\). 2. Recall the su

1. **Problemstellung:** Wir sollen die Gleichungen aus Aufgabe 3 lösen und die Lösungen unter der Waage angeben. 2. **Formeln und Regeln:**

1. The problem states: Given $a - b = \frac{14}{3}$, find $a^2 - b^2$. 2. Recall the algebraic identity: $$a^2 - b^2 = (a - b)(a + b)$$

1. **State the problem:** Solve the equation $\frac{y}{x} = 1.1$ for $y$ in terms of $x$. 2. **Formula and rules:** The equation $\frac{y}{x} = 1.1$ means that $y$ divided by $x$ e

1. **State the problem:** We are given the equation $\frac{y}{x} = 6.6$ and need to express $y$ in terms of $x$. 2. **Formula and rules:** The equation shows a ratio of $y$ to $x$.

1. The problem asks us to simplify the expression $16^{\frac{3}{4}}$. 2. Recall the rule: the $n$-th root of a number $a$ is $a^{\frac{1}{n}}$. Therefore, $a^{\frac{m}{n}} = \left(

1. The problem asks to simplify the expression $16^{\frac{3}{4}}$. 2. Recall the rule: the $n$-th root of a number $a$ is $a^{\frac{1}{n}}$. Therefore, $a^{\frac{m}{n}} = \left(a^{