🧮 algebra

1. **Enunțul problemei:** Avem un număr natural $n$ care, împărțit la 6, 8, 9, 12 și 16, dă același rest 5. 2. **Ce înseamnă asta?** Dacă împărțim $n$ la fiecare dintre aceste nume

1. **State the problem:** We are given a line $2y=2x+29$ and a quadratic equation $x^2 - 2ky + 32k = 0$ where $k \neq 0$. The line is tangent to the quadratic, and we need to find

1. **State the problem:** Simplify the expression $$\sqrt{9 + 4\sqrt{5}}$$. 2. **Recall the formula:** Expressions of the form $$\sqrt{a + b}$$ where $$b$$ involves a square root c

1. Το πρόβλημα ζητά να εκφράσουμε το $\sqrt{a + b\sqrt{c}}$ με απλούστερη μορφή. 2. Υποθέτουμε ότι υπάρχει μια μορφή $\sqrt{a + b\sqrt{c}} = \sqrt{x} + \sqrt{y}$, όπου $x$ και $y$

1. **Problem 4:** Find $f(-8)$ when $f(x) = -2x + 3$. 2. **Formula:** To find $f(a)$, substitute $x = a$ into the function: $$f(a) = -2a + 3$$

1. **Problem:** Find the product $f(n) \cdot g(n)$ where $f(n) = -3n^2 - 2$ and $g(n) = 4n - 5$. 2. **Formula:** To multiply two polynomials, use the distributive property: $(a+b)(

1. **Problem Statement:** Find the composite functions $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$ for $f(x) = 2x - 1$ and $g(x) = x^2$. Also, state the domain of each c

1. **State the problem:** We are given two functions:

1. **State the problem:** We are given two functions $$f(x) = 2\sqrt{x}$$ and $$g(x) = 7x$$ and need to evaluate the compositions: (a) $$(f \circ g)(4)$$

1. **State the problem:** We are given two functions $f(x) = 4x$ and $g(x) = 8x^2 + 5$. We need to evaluate the compositions: (a) $(f \circ g)(4)$ means $f(g(4))$

1. **State the problem:** We need to evaluate the composite functions $(g \circ f)(x)$ and $(f \circ g)(x)$ at given values of $x$. This means: - $(g \circ f)(x) = g(f(x))$

1. **State the problem:** We need to evaluate composite functions such as $(f \circ g)(1)$, $(f \circ g)(-1)$, $(g \circ f)(-1)$, $(g \circ f)(0)$, $(g \circ g)(-2)$, and $(f \circ

1. **State the problem:** We need to find the domain of the function $$f(x) = \frac{x^2 - 25}{x^2 - 16}$$. 2. **Recall the domain rule for rational functions:** The domain includes

1. **State the problem:** We are given a table of temperatures in Fahrenheit ($F$) and Celsius ($C$) below freezing and asked to complete the equation in standard form relating $F$

1. **State the problem:** We are given the line equation $$y - 11 = -\frac{4}{3} (x + 7)$$ and need to find which of the given equations represent the same line. 2. **Start by dist

1. **Problem 46:** Evaluate $0.64^{\frac{7}{2}}$ and $0.64^5$. 2. **Step 1:** Express $0.64$ as a fraction: $0.64 = \frac{64}{100} = \frac{16}{25}$.

1. **State the problem:** Solve the inequality $$3^{1 - x^2} - \frac{1}{9} < 0$$. 2. **Rewrite the inequality:** Move $$\frac{1}{9}$$ to the other side:

1. **State the problem:** Solve the inequality $$5^{x+2} + 5^{x+1} < 5^{2x} + 125$$. 2. **Rewrite the terms using properties of exponents:**

1. **State the problem:** Solve the equation $b = 12b + 1$ for $b$. 2. **Rewrite the equation:**

1. **State the problem:** Solve the equation $b = \frac{12}{b} + 1$ for $b$. 2. **Rewrite the equation:**

1. **State the problem:** Calculate the value of $99999999\sqrt{9292929922929}$.\n\n2. **Recall the formula:** The expression involves a multiplication of a number and the square r