🧮 algebra

1. **Solve for x in the equation 4x - 9 = -21.** 2. Start by isolating the term with x. Add 9 to both sides:

1. **Problem:** Calculate the value of the expression $$\frac{\sqrt{2}+1}{\sqrt{3}+1} + \frac{\sqrt{2}-1}{\sqrt{3}-1}$$

1. **State the problem:** Simplify the expression $\frac{1}{1}$. 2. **Recall the rule:** Any number divided by itself (except zero) equals 1.

1. **State the problem:** We are given two functions: $m(x) = \sqrt{x - 4}$ and $n(x) = x + 1$. We need to find the domain and range of the composition $(m \circ n)(x) = m(n(x))$.

1. The problem asks to find 60% of 355. 2. To find a percentage of a number, use the formula: $$\text{Percentage of a number} = \frac{\text{percentage}}{100} \times \text{number}$$

1. **State the problem:** Find 30% of 245. 2. **Formula:** To find a percentage of a number, use the formula:

1. **State the problem:** We are given two functions $h(x) = \frac{1}{2x - 8}$ and $g(x) = -2x$. We need to find the domain and range of the composition $(h \circ g)(x) = h(g(x))$.

1. **State the problem:** We need to resolve the given rational function into partial fractions. 2. **General formula and rules:** For a rational function $\frac{P(x)}{Q(x)}$ where

1. Stating the problem: We want to decompose the rational expression $$\frac{5x+1}{(2x+1)^2}$$ into partial fractions of the form $$\frac{A}{2x+1} + \frac{B}{(2x+1)^2}$$. 2. Formul

1. **State the problem:** We have a geometric progression (GP) where the sum of the 2nd and 5th terms is 72, and the sum of the 3rd and 6th terms is 144. We need to find the common

1. **Problem statement:** Convert the given expression into partial fractions and find the value of $x$. 2. **General formula and rules:** Partial fraction decomposition is used to

1. **State the problem:** Simplify or analyze the expression $$\frac{5x+1}{(2x+1)^2}$$. 2. **Formula and rules:** This is a rational expression where the numerator is a linear poly

1. Planteamos el problema: calcular el valor de $$E = \frac{-4(3) + 7(-2) + 20}{-5 + 2(-8) + (-3)(4)}$$. 2. Evaluamos cada término en el numerador:

1. **State the problem:** Simplify the expression $\frac{6x+9}{x}$. 2. **Recall the formula and rules:** When dividing a sum by a term, you can split the division over each term: $

1. **State the problem:** A school has $5x + 10$ books shared equally among $x$ students. We need to write the rational expression for books per student and then simplify it. 2. **

1. **State the problem:** We need to find the sum of the first 5 terms and the nth term of the geometric series 5 + 15 + 45 + 135 + ... 2. **Identify the series type and formula:**

1. Problem: Find the roots of the quadratic equation $$3x^2 - 2x - 5 = 0$$ 2. Formula: The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by the quadratic formula:

1. **State the problem:** We need to find the sum of the first 5 terms and the nth term of the geometric series 4, 8, 16, 32, ... 2. **Identify the series type and formula:** This

1. **State the problem:** Calculate the value of the expression $$E = \frac{-4(3) + 7(-2) + 20}{-5 + 2(-8) + (-3)(4)}$$. 2. **Apply multiplication in numerator and denominator:**

1. **State the problem:** We are given the equation $x^3 + y^3 = 20$ and want to understand or analyze it. 2. **Formula and rules:** This is an equation involving cubes of two vari

1. **State the problem:** Evaluate the expression $$E = \frac{-4(3) + 7(-2) + 20}{-5 + 2(-8) + (-3)(4)}$$. 2. **Apply multiplication:** Calculate each product inside the numerator