🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{1}{2} \sqrt{2} - \sqrt{2}$$. 2. **Recall the rules:** When subtracting like terms involving square roots, treat the square

1. The problem is to understand the relationship between $-\sqrt{2}$ and $\sqrt{2}$. 2. Recall that $\sqrt{2}$ is the positive square root of 2, approximately 1.414.

1. The problem is to find the decimal value of the expression $\frac{1}{2} \sqrt{2} - \sqrt{2}$. 2. Recall that $\sqrt{2}$ is the square root of 2, approximately equal to 1.414.

1. **Problem:** Evaluate the expression without using a calculator or mathematical table: $$\frac{5}{6} \text{ of } (4 \times 1 \times 3 \times 5 \times 5 \times 6) \times \frac{5}

1. **State the problem:** Simplify the expression $$-27 + 18 - (10 - 14) - (-2)$$. 2. **Recall the rules:**

1. **State the problem:** We are given the product of three numbers is 1197, and two of the numbers are 3 and 19. We need to find the third number. 2. **Formula and rule:** The pro

1. **State the problem:** Simplify and substitute values into the expression $c = 4(x+2y) - 2x$ to find the total cost, where $x$ is the LCM of 6 and 8, and $y$ is the smallest cub

1. **Stating the problem:** We are given a sequence with terms corresponding to positions $n=100, 101, 102$ having values $412, 416, 420$ respectively. We want to find the general

1. **State the problem:** We are given the quadratic sequence: -5, -2, 3, 10, 19, ... and need to find the formula for the $n^{th}$ term. 2. **Recall the formula for a quadratic se

1. **State the problem:** We are given the quadratic sequence: -5, -2, 3, 10, 19, ... and need to find the formula for the $n^{th}$ term. 2. **Identify the pattern:** Quadratic seq

1. **State the problem:** We are given the equation of two lines combined as $$x^2(\tan^2\theta + \cos^2\theta) - 2xy \tan\theta + y^2 \sin^2\theta = 0$$ and need to show that the

1. **Problem statement:** Show that $x+2$ is a factor of the polynomial $p(x)$. 2. **Key concept:** A polynomial $p(x)$ has a factor $x - a$ if and only if $p(a) = 0$. This is know

1. **State the problem:** We need to find the remainder when the polynomial $p(x) = 15x^3 + 22x^2 - 15x + 2$ is divided by $x + 1$. 2. **Recall the Remainder Theorem:** The remaind

1. **State the problem:** We are given a cubic polynomial $p(x) = 2x^3 + ax^2 + bx + c$ with roots $x=\frac{1}{2}$, $x=n$, and $x=-n$, where $a,b,c,n$ are integers. The $y$-interce

1. **State the problem:** Simplify the expression $\left(3x^{-2} y\right)^2$. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to tha

1. **Problem statement:** There are 25 students who took math and science exams. 17 passed science, 8 passed both math and science, and 3 did not pass either exam. We need to find

1. **Problem:** Simplify $9^{3/2}$. 2. **Formula and rules:** For any positive number $a$ and rational exponent $m/n$, we use the rule:

1. समस्या: भिन्नों को मिश्र भिन्न में बदलना। 2. मिश्र भिन्न में बदलने का सूत्र: यदि भिन्न $$\frac{a}{b}$$ है, जहाँ $$a > b$$, तो इसे मिश्र भिन्न के रूप में लिखा जाता है:

1. **Problem Statement:** Given the total cost of 3 mangoes and 4 apples is 160, and after increasing the cost of a mango by 20% and decreasing the cost of an apple by 20%, the tot

1. **State the problem:** We need to find the points of intersection A and B between the line $y = 5x + 6$ and the curve $xy = 8$. 2. **Write down the equations:**

1. **State the problem:** We need to find the equation of a water pipe line whose gradient is the reciprocal of a given prime number. The line must pass through the point $P(a,b)$,