🧮 algebra

1. Problem 17: Find $k$ if $k+7$, $2k-2$, and $2k+6$ are three consecutive terms of an AP. Step 1: In an arithmetic progression (AP), the difference between consecutive terms is co

1. **Problem:** If the zeroes of the quadratic polynomial $ax^2 + bx + \frac{2a}{b}$ are reciprocal of each other, find the value of $b$. Step 1: Let the roots be $\alpha$ and $\be

1. Problem: Two cars A and B take 30 minutes and $p$ minutes respectively to complete 1 round. They meet first time after 90 minutes at the starting point.

1. **State the problem:** Solve the system of linear equations: $$x+y+z=5$$

1. We are given the system of equations representing two circles: $$ (x-10)^2 + (y-5)^2 = 25 $$

1. **نص المشكلة:** لدينا عددان \( a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5}} \) و \( b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5}} \). المطلوب هو كتابة كل منهما على شكل نسبة مقاماها عدد ن

1. نبدأ بتوضيح قيم $a$ و $b$: $$a = \frac{\sqrt{5} - 1}{\sqrt{5}}, \quad b = \frac{\sqrt{5} - 1}{\sqrt{5}}$$

1. **State the problem:** We are given the system of equations: $$2x + y + z = 6$$

1. Let's analyze the given series: 2, 5, 9, 19, 37, ___. 2. Find the differences between consecutive terms:

1. The problem is to simplify the expression $$\sqrt{a+n} - \sqrt{a-n}$$. 2. To simplify, we multiply and divide by the conjugate of the expression to rationalize it:

1. Determine $\Delta = a_{11}a_{22} - a_{12}a_{21}$ for system \(\begin{cases} x + y = 3 \\ x + 2y = -8 \end{cases}\). Coefficients: $a_{11}=1$, $a_{12}=1$, $a_{21}=1$, $a_{22}=2$

1. First, state the problem: Solve the equation $$3 \cdot 9^{x^2+1} - 3^{x^2+2} + 18 = 0$$ for $x$. 2. Rewrite the bases to express all terms as powers of 3, since $9 = 3^2$:

1. The problem provides 14 pairs of linear equations with variables $x$ and $y$. 2. For each pair, solve the system using substitution or elimination.

1. Let's analyze the given series: 7, 20, 58, 171, ___. 2. We look for a pattern or rule that connects the terms.

1. **State the problem:** We are given two stages of a journey, each covering 30 km. The average speeds are $a$ km/h for the first stage and $b$ km/h for the second stage. We want

1. Let's analyze the series: 2, 7, 14, 23, 34, 47, 62, ___? \n 2. First, find the differences between consecutive terms: \n

1. **State the problem:** We have three consecutive positive integers with $n$ as the middle integer. Let the three integers be $(n-1)$, $n$, and $(n+1)$. We multiply these three i

1. Problem 1: Solve the system: $$\frac{4}{5}x - 4 + y + 1 = 1$$

1. **State the problem:** Solve the equation $$2^{x+1} - 11 + \frac{15}{2^x + 1} = 0$$ for $x$. 2. **Rewrite and simplify:** Note that $$2^{x+1} = 2 \cdot 2^x$$. Let $y = 2^x$. The

1. The problem is to convert the decimal number 86.52 into a decimal fraction. 2. First, write the decimal number as a fraction with 1 as the denominator: $86.52 = \frac{86.52}{1}$

1. The problem asks to convert 58.34 to a decimal number. 2. The given number 58.34 is already in decimal form because it has a decimal point.