🧮 algebra

1. **State the problem:** We have a rectangle with an original length of 8 cm and an area of 20 cm². We want to find the original width, then analyze what happens when the length i

1. **State the problem:** We have a stove with a cash price of 800.

1. Let's clarify the problem: You mentioned the answer to the first question was 6, 8, 10. 2. These numbers (6, 8, 10) form a Pythagorean triple, which means they satisfy the Pytha

1. The problem is to find the pattern or rule for the sequence 6, 8, 10. 2. Observe the differences between consecutive terms: $8 - 6 = 2$ and $10 - 8 = 2$.

1. The problem asks for the multiplication inverse of the fraction $\frac{a}{b}$. 2. The multiplication inverse (also called the reciprocal) of a number is the number which, when m

1. The problem asks to find 33 and a third percent of 100. 2. First, convert 33 and a third percent to a fraction or decimal. 33 and a third percent is equal to $33\frac{1}{3}\% =

1. **State the problem:** We have a club with $x$ adults. Out of these, some are over 30 years old.

1. **Calculate the sum and difference of the given numbers:** Given expression:

1. **Problem statement:** Given positive numbers $a, b, c$ such that $a + b + c = 18$, prove that $$\frac{a}{b^2 + 36} + \frac{b}{c^2 + 36} + \frac{c}{a^2 + 36} \geq \frac{1}{4}.$$

1. The problem asks to complete the sentence: "It is not possible to divide 22 by 0 because there is no number you can _________ by 0 and get 22." 2. To understand this, recall the

1. **State the problem:** Given functions $f(x) = \sqrt{x}$ and $g(x) = x^3 + 2$, find the compositions $f \circ g$, $g \circ f$, $f \circ f$, $g \circ g$ and their domains. 2. **F

1. **State the problem:** We need to find the value of $T = \frac{m}{f}$ given the ranges for $m$ and $f$. 2. **Given data:**

1. **State the problem:** Solve the equation $$2x + \frac{1}{x} = 3y - \frac{2}{y} - 5$$ for one variable in terms of the other. 2. **Rewrite the equation:** Move all terms to one

1. **State the problem:** Factor the expression $$g^2 - h^2 - 10h - 25$$ completely. 2. **Rewrite the expression:** Group the terms involving $h$ together:

1. The problem is to adjust the number 3 by putting the denominator of the term \frac{3y-2}{y}. 2. The term given is \frac{3y-2}{y}, where the denominator is $y$.

1. **Problem 1:** If the roots of the quadratic equation $ax^2 + cx + c = 0$ are in the ratio $p : q$, prove that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0

1. Let's start by understanding what special products are. Special products are shortcuts to multiply certain types of algebraic expressions quickly and easily. 2. One common speci

1. **State the problem:** Simplify the expression $$\frac{x^2 - y^2 + 3x + 3y}{x - y + 3}$$ to its lowest terms. 2. **Factor the numerator:** Notice that $$x^2 - y^2$$ is a differe

1. Simplify $\frac{x - 2}{6x} + \frac{2x + 1}{3x}$. Find common denominator: $6x$.

1. **State the problem:** Factor the cubic polynomial $$m^3 + m^2 - 4m - 4$$ completely. 2. **Group terms:** Group the polynomial into two pairs to factor by grouping:

1. **State the problem:** Factor completely the expression $$(x + 5)^2 - 81$$. 2. **Recognize the form:** This expression is a difference of squares because it can be written as $$