🧮 algebra

1. **State the problem:** We have points $P(-5,a)$ and $Q(7,3a)$ on a coordinate plane, with $a>0$. The length of segment $PQ$ is given as $4\sqrt{10}$. We need to find the equatio

1. **State the problem:** Solve the quadratic equation $$x^2 + 20x - 300 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the e

1. Állítsuk fel a feladatot: Oldjuk meg az egyenletet $$\frac{2x+3}{x-1}=4$$. 2. Az egyenlet megoldásához szorozzuk meg mindkét oldalt az $x-1$ nevezővel, hogy megszabaduljunk a tö

1. **State the problem:** Factor the expression given by the user. Since no specific expression was provided, let's consider a general example: factor $x^2 - 5x + 6$. 2. **Formula

1. **State the problem:** Simplify the expression $$4(m+2n)+9(3m+6n)-12(5m+10n)$$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor o

1. Let's clarify the problem: You are asking why the expression $d/10$ appears in step 3 of a solution. 2. Typically, $d/10$ means dividing the variable $d$ by 10, which could repr

1. **Problem statement:** Ravi starts for school at 8:20 a.m. traveling at 10 km/h and arrives 8 minutes late. Traveling at 16 km/h, he arrives 10 minutes early. We need to find th

1. **Problem statement:** The water supply in a hotel is enough for all guests for 8 days. We need to find how many days the water supply will last if only 40% of the guests are pr

1. **State the problem:** We are given three angles of a triangle: $3h$, $h+10$, and $2h-40$. We need to find the smallest angle and compare it to 35. 2. **Recall the triangle angl

1. The problem is to find the common ratio $r$ in a geometric series given the sum formula $S_n = \frac{a_1}{1-r}$. 2. The formula $S_n = \frac{a_1}{1-r}$ is used for the sum of an

1. The problem is to find the sum of the series: $\frac{1}{6} + \frac{1}{2} + \frac{3}{2} + \ldots$ 2. First, identify the pattern or rule for the terms in the series.

1. **Problem:** 81 is 73% of what number? 2. **Formula:** To find the base number when given a part and a percentage, use the formula:

1. **State the problem:** We need to find the number $x$ such that 81 is 78% of $x$. 2. **Write the equation:** Since 78% means $\frac{78}{100}$, the equation is:

1. **State the problem:** We have total students = 4700 across institutes U, V, W, X, Y, Z with given percentages. 2. **Calculate total students per institute:**

1. **Problem Statement:** Two trains start from stations 200 miles apart at 10:00 am, traveling towards each other at 100 mph each. A vulture flies back and forth between the train

1. **State the problem:** Solve the equation $$\frac{1}{a} + \frac{1}{b} = 2$$ for one variable in terms of the other. 2. **Formula and rules:** To combine fractions, find a common

1. **State the problem:** We need to find the percentage of boys studying in the school in 1998 relative to the number of girls studying in the school in 1999.

1. The problem is to find the value of $20260104100050567$ given the values of $20260104100050566 = 7$ and $20260104100050565 = 8$. 2. Since the problem does not specify a direct r

1. **Menyatakan masalah:** Kita diminta untuk mengembangkan dan memahami fungsi $f(x) = \left(\frac{1}{2}x^2 - 1\right)(4x + 3)$.\n\n2. **Rumus yang digunakan:** Untuk mengalikan d

1. نبدأ ببيان المشكلة: لدينا كميات أ، ب، ج، د متناسبة، ونريد إثبات أن $$\frac{أ}{ب} - أ = \frac{ج}{د} - ج$$. 2. بما أن الكميات أ، ب، ج، د متناسبة، فهذا يعني أن هناك ثابت تناسب \(k\

1. **Problem:** Find the length of a side of a square whose perimeter is 1600 cm. 2. **Formula:** The perimeter $P$ of a square is given by