🧮 algebra

1. **State the problem:** Simplify and solve the expression $3x(3x+3)$. 2. **Use the distributive property:** Multiply $3x$ by each term inside the parentheses:

1. **State the problem:** Simplify or analyze the expression $$(0.4\log(x) + 0.88)^2$$ where $\log$ is the logarithm base 10. 2. **Recall the formula:** The square of a binomial $(

1. We start with the first problem: find for which values of $x$ the graph of $f(x)$ lies below the graph of $g(x)$. 2. For (a), the functions are $f(x) = x^2$ and $g(x) = -4x + 5$

1. **State the problem:** Express the rational function $$\frac{x^2 + 4}{x^3 + 2x}$$ in partial fractions. 2. **Factor the denominator:**

1. **State the problem:** Solve the system of equations: $$x + 2y = 1$$

1. **State the problem:** We want to express the rational function $$\frac{x - 2}{x^2 - 2x + 1}$$ in partial fractions.

1. Planteamos el problema: Encontrar la función inversa de $$f(x) = \frac{2x}{x-1}$$. 2. Para hallar la función inversa, intercambiamos $$x$$ y $$y$$ en la ecuación y despejamos $$

1. Das Problem lautet: Faktorisiere die Gleichung $$0=5x^4-198x^2+800$$. 2. Wir erkennen, dass es sich um ein Polynom vierten Grades handelt, das nur gerade Potenzen von $x$ enthäl

1. **State the problem:** We have two functions:

1. The problem is to solve the equation given by the user. However, no specific equation was provided, so I will demonstrate solving a simple algebraic equation as an example: $2x

1. Das Problem lautet: Finde die Nullstellen der Funktion $5x^4 - 198x^2 + 800 = 0$ durch Faktorisieren. 2. Wir verwenden die Substitution $y = x^2$, damit die Gleichung zu einer q

1. The problem is to find the value of $c$ given some context where the answer key states $c=20$. 2. Since the user only mentioned the answer key's value for $c$, we need to verify

1. **State the problem:** We have a rectangle QRST with sides QR = $x$, QT = $3x - 10$, RS = $2y$, and TS = $4y$. Since QRST is a rectangle, opposite sides are equal in length. 2.

1. **State the problem:** Multiply the fraction $\frac{4}{6}$ by the mixed number $3 \frac{3}{5}$. 2. **Convert the mixed number to an improper fraction:**

1. The problem asks to write each expression as a single power of 2. 2. Recall the exponent rules:

1. The problem is to understand the variable $x$ and its role in algebra. 2. In algebra, $x$ is commonly used as a variable representing an unknown value.

1. **State the problem:** Write the expressions \(2 \times 2^a\) and \(4 \times 2^b\) as a single power of 2. 2. **Recall the rule for multiplying powers with the same base:**

1. We are asked to solve a quadratic equation by completing the square. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$.

1. **State the problem:** Solve the quadratic equation $x^2 - 2x - 1 = 0$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are gi

1. **State the problem:** Solve the equation $2^3 = 2^5$. 2. **Recall the rule for exponents:** If $a^m = a^n$ and $a \neq 0$, then $m = n$.

1. **State the problem:** Solve the equation $2^{3x+1} = 4$ for $x$. 2. **Recall the formula and rules:** We know that $4$ can be written as a power of $2$, since $4 = 2^2$.