🧮 algebra

1. Planteamos el problema: Pedro divide su terreno en dos partes, llamémoslas $x$ y $y$, con $x + y = 11600$ m². 2. Según el enunciado, $\frac{2}{5}$ de la primera parte mide lo mi

1. **Problem Statement:** Learn how to work with exponents, including evaluating expressions with positive and negative exponents, simplifying expressions with exponents, and apply

1. **State the problem:** Solve the equation $$x^4 - 5x^2 + 4 = 0$$ for $x$. 2. **Use substitution:** Let $y = x^2$. Then the equation becomes a quadratic in $y$:

1. **Сформулюємо першу систему рівнянь:** $$\begin{cases} x + y = 5 \\ y^2 + 4xy = 33 \end{cases}$$

1. **State the problem:** Find the non-zero values of $a$ and $b$ such that $$(2x - a)^3 = 8x^3 - bx^2 + \frac{3}{2} b x - a^3.$$

1. **State the problem:** Find the non-zero values of $a$ and $b$ such that $$(2x - a)^3 = 8x^3 - bx^2 + 2bx - a^3.$$ 2. **Recall the binomial expansion formula:** For any $u$ and

1. **Problem:** Simplify the expression $$\frac{\frac{3y^2 + 9x^2 - 7a - 9}{7x - 9}}{\frac{x^2 - 3x + 11}{3x - 9 - 11}}$$. 2. **Formula and rules:** When dividing fractions, multip

1. **State the problem:** Solve the system of inequalities: $$\frac{x-2}{x-4} < 0$$

1. **State the problem:** Find all real solutions to the equations: (i) $$16a^2 = 2\sqrt{a}$$

1. The problem states that the roots of the cubic equation $$x^3 - 2x^2 + 4x - 5 = 0$$ are $$p$$, $$q$$, and $$r$$. We need to find the value of $$(p - 4)(q - 4)(r - 4)$$ without s

1. **State the problem:** We need to determine which expressions always yield an even value when $x$ is an odd number. 2. **Recall important rules:**

1. **State the problem:** We need to determine which expressions always yield an even value when $x$ is an odd number. 2. **Recall important rules:**

1. **State the problem:** We are given the equation $$f = \frac{b^2 + w}{2 - b}$$ and need to find the value of $w$ when $b = 4$ and $f = -10$. 2. **Write down the formula:** $$f =

1. **State the problem:** We have a system of three equations: $$kx + 3y - z = 3$$

1. **State the problem:** We need to find the value of $w^r$ when $w=5$ and $r=3$. 2. **Formula:** The expression $w^r$ means $w$ raised to the power of $r$, which is multiplying $

1. **State the problem:** We need to find the value of $6 + y^2$ when $y = 3$. 2. **Formula used:** The expression is $6 + y^2$.

1. **State the problem:** Solve for $x$ in the equation $4x = 13x - 18$. 2. **Rewrite the equation:** Move all terms involving $x$ to one side to isolate $x$.

1. **State the problem:** Solve the equation $4x - 13x = -38$ for $x$. 2. **Combine like terms:** On the left side, combine $4x$ and $-13x$.

1. The problem is to solve the inequality $\ln x \geq -1$. 2. Recall that the natural logarithm function $\ln x$ is defined only for $x > 0$.

1. **State the problem:** Simplify or factor the polynomial $2x^3 - 3x^2 - 3x + 2$. 2. **Recall the factoring approach:** For cubic polynomials, try to find rational roots using th

1. **State the problem:** We are comparing the slopes of two linear functions, Function A and Function B. 2. **Given information:**