🧮 algebra

1. Vamos resolver um problema fácil de álgebra: Resolver a equação $2x + 3 = 7$. 2. A fórmula básica para resolver equações lineares é isolar a variável $x$.

1. **State the problem:** We have a kite with four sides: two sides each of length $n$ cm and two sides each of length $n + 3$ cm. The perimeter is given as 102 cm. 2. **Formula fo

1. **State the problem:** Solve for $a$ in the equation $$\left(\frac{1}{9}\right)^{a+1} = 81^{8+a+1} \cdot 27^{2 - a}.$$\n\n2. **Rewrite bases as powers of 3:**\n- $9 = 3^2$, so $

1. **State the problem:** Solve the equation $$\left(\frac{1}{9}\right)^{a+1} = 81^{8+a+1} \cdot 27^{2-a}$$ for the variable $a$. 2. **Rewrite bases as powers of 3:**

1. The problem asks for the area of a square with side length $4x + 3$. 2. The formula for the area of a square is:

1. **State the problem:** A rectangular picture frame measures 20 cm by 30 cm. A mat of uniform width $x$ cm is placed inside the frame, creating a border. The area of the mat is e

1. **Problem statement:** We need to sketch a quartic polynomial function $y = f(x)$ such that $f(x) > 0$ when $x < -5$, $2 < x < 3$, and $x > 4$. We also need to write an inequali

1. **State the problem:** Lilly thinks of a number $k$. She triples it and then subtracts 8 to get 7. 2. **Write the equation:** Tripling $k$ means $3k$. Subtracting 8 gives $3k -

1. **State the problem:** We are given the expression $y = (a + b)(a - b)$ and need to find the value of $y$ when $a = 7.5$ and $b = 2.5$.

1. **State the problem:** We need to find the value of $y$ given the expression $y = (a + b)(a - b)$ when $a = 7.5$ and $b = 2.5$. 2. **Recall the formula:** The expression $(a + b

1. Add the polynomials. **a)** $(3x^2 - x + 2) + (4x^2 + 3x - 1)$

1. **State the problem:** Solve the quadratic equation $$2x^2 + 9x - 5 = 0$$ for $x$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where

1. The problem is to find the value of $9^{\frac{3}{4}}$. 2. Recall the rule for fractional exponents: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$.

1. **State the problem:** Solve the equation $$\frac{x}{x-1} + \frac{x}{x+1} - \frac{3}{x-1} = 4$$ using the quadratic formula. 2. **Rewrite the equation:** Move all terms to one s

1. **State the problem:** Evaluate the expression $1^3 + 4 \cdot 1^2 - 11 \cdot 1 - 30$. 2. **Recall the order of operations:** Calculate exponents first, then multiplication, and

1. **State the problem:** Solve the equation $$\Delta an(2x - \pi) = \frac{\sqrt{2}}{5}$$ for $$x$$. 2. **Recall the definition and properties:** The function $$\Delta an$$ is the

1. **State the problem:** Factor the quadratic expression $7x^2 + 14x - 105$ and find the values of $x$ that satisfy $7x^2 + 14x - 105 = 0$. 2. **Write the equation:**

1. El problema nos pide escribir una ecuación que represente el valor de un iPad que cuesta 700 y se deprecia un 20% cada año. 2. La fórmula para depreciación exponencial es:

1. We are asked to simplify the expression $$\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2 + \sqrt{3}}}$$. 2. To simplify a fraction with square roots in numerator and denominator, we can mul

1. **State the problem:** Simplify and analyze the quadratic expression $-4x^2 - 32x - 28$. 2. **Formula and rules:** To simplify, factor out the greatest common factor (GCF) from

1. The problem asks for the end behavior of the polynomial function $$f(x) = -7x^6 - 63x^5 - 77x^4 + 343x^3 + 168x^2 - 700x + 336$$. 2. To determine end behavior, focus on the term