🧮 algebra

1. **Problem Statement:** Show that if $\alpha$ and $\beta$ are roots of the quadratic equation $$x^2 - 14x + 36 = 0,$$ then for all positive integers $n$, the expression $$\alpha^

1. **State the problem:** We are given points on an exponential function: $(-1, \frac{1}{7})$, $(0, 1)$, $(1, 7)$, and $(2, 49)$. We want to show that the function grows by equal f

1. **Problem Statement:** Given a parabola that opens upward, crosses the x-axis near -5 and -3, has a vertex near (-4, -9), and does not cross the y-axis.

1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** The zero product property states that if a product of two factors equals zer

1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** If a product of two factors equals zero, then at least one of the factors mu

1. Problemet: Vi skal finde den afledte af funktionen $$f(x) = \ln(x) \cdot (5x^4 + 2)$$ og derefter bestemme $$f'(1)$$. 2. Formel: Vi bruger produktreglen for differentiation, som

1. The problem asks to find $f(g(3))$ given the functions $f(x) = 3x - 7$ and the graph of $g$. 2. From the graph, find the value of $g(3)$. Observing the graph, at $x=3$, $g(3) =

1. **Enunciare il problema:** Discutere e risolvere il sistema lineare a parametri reali $k$:

1. **State the problem:** We want to find an approximate solution to the equation $$x^3 - 4x + 1 = 0$$ using the iterative formula $$x_{n+1} = - \frac{1}{x_n^2 - 4}$$ starting with

1. **State the problem:** We want to find an approximate solution to the equation $$x^3 - 5x^2 - 12 = 0$$ using the iterative formula $$x_{n+1} = 5 + \frac{12}{x_n^2}$$ starting wi

1. **State the problem:** We need to find the discount percentage when the original price of a couch is 405 and the discounted price is 162.

1. **State the problem:** A store owner discounted some rare acoustic guitars from $7,325 to $2,344. We need to find the discount percentage. 2. **Formula for discount percentage:*

1. **Problem Statement:** We need to write an equation for a polynomial function given its graph.

1. **State the problem:** Expand and simplify the expression $ (6x + 7)^2 $. 2. **Formula used:** The square of a binomial $ (a + b)^2 = a^2 + 2ab + b^2 $.

1. **State the problem:** Solve the equation $$\frac{x^2 + x - 30}{x - 5} = 1$$ for $x$. 2. **Recall the formula and rules:** To solve rational equations, multiply both sides by th

1. **State the problem:** Simplify the expression $32x - 11 \times 3x$. 2. **Recall the order of operations:** Multiplication must be done before subtraction.

1. **State the problem:** Simplify and analyze the expression $3^x \times (3^x - 11)$. 2. **Recall the formula:** When multiplying terms with the same base, use the property $a^m \

1. Das Problem lautet: Löse die Gleichung $$\frac{2x+4}{3} = 6$$. 2. Die Formel, die wir verwenden, ist das Lösen von Gleichungen durch Multiplikation beider Seiten mit dem Nenner,

1. The problem is to solve the equation or expression given previously, but using a different method as requested. 2. Since the original problem is not restated, let's assume it in

1. **State the problem:** Write the quadratic equation $$y = -\frac{7}{4}x^2 - \frac{1}{2}x + 2$$ in vertex form, and identify the vertex, direction of opening, and y-intercept. 2.

1. **State the problem:** Solve the equation $$\frac{2x+4}{3} = 5$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation. We can