🧮 algebra

1. **Problem 1: Find the general equation of the hyperbola with vertices (0, ±2) and foci (0, ±2\sqrt{5})**. Step 1: Identify the orientation and parameters.

1. **State the problem:** We want to simplify the expression $$\log \left( \sqrt{\frac{7^2 t^3 p}{d^6 b^2}} \right)$$. 2. **Rewrite the square root as an exponent:** Recall that $$

1. We are given the equation $x + \frac{1}{x} = 3$ and need to find the value of $x^{15} + \frac{1}{x^{15}}$. 2. Let's denote $a_n = x^n + \frac{1}{x^n}$. We know $a_1 = 3$.

1. Problem: Simplify the expression $$\log\left(\sqrt{\frac{7^2 t^3 p}{d^t b^2}}\right)$$. 2. First, rewrite the square root as a fractional exponent: $$\sqrt{x} = x^{1/2}$$, so

1. Simplify the expressions. 1.a. Simplify $3(x - y) - 3(2x + 3y)$

1. **State the problem:** We have a rectangular garden that is 600 m long and 300 m wide, and we need to calculate the cost of fencing around it. 2. **Calculate the perimeter of th

1. The problem is to solve the equation $2 - 6 = \frac{\boxed{\quad}}{3}$, where the numerator of the right side fraction is unknown. 2. Simplify the left side: $2 - 6 = -4$.

1. The problem is to find the missing numerator in the fraction \( \frac{\_}{2} \) such that \( 4 - 8 = \frac{\_}{2} \). 2. First, calculate the left side: \(4 - 8 = -4\).

1. **State the problem:** We are given a rectangle with a perimeter of 60 m and one side measuring 18 m. We need to find the area of the rectangle.

1. **State the problem:** Illustrate a real-life example showing inverse variation. A common example is the relationship between the speed of a car and the travel time for a fixed

1. The problem asks for the simplest form of the expression $4 + 4i$. 2. The expression $4 + 4i$ cannot be simplified further by combining like terms since the real part is $4$ and

1. **State the problem:** We have a function $f(n) = \frac{3x^3 + b}{6x}$ and its inverse function \(f^{-1}(x) = a x + c \sqrt{x^2 + 1}\). We are asked to find $a + b + c$.

1. The user provided expressions: $f(x) = xr^2 + b$

1. مسئله را بیان کنیم: تابع \( f(x) = ax^2 + bx + c \) است و معکوس آن به صورت \( f^{-1}(x) = ax + c\sqrt{xr + 1} \) داده شده است. قصد داریم مقدار \( r \) را پیدا کنیم. 2. تابع وارو

1. The problem asks for the equation of the axis of symmetry of a parabola. 2. The graph shows a parabola opening downwards with its vertex at approximately (0, 5).

1. The problem states we have a function $$f(n) = \frac{3x^3 + b}{6x}$$ and its inverse $$f^{-1}(x) = a x + c \sqrt{x^2 + 1}$$. We are asked to find the value of $$a + b + c$$. 2.

1. The problem is to evaluate the expression $\frac{(-2)^2}{3^2} \times (-6)$. 2. First, calculate $(-2)^2$. Squaring a negative number results in a positive number, so $(-2)^2 = 4

1. **Stating the problem:** We are given a polynomial graph $p$ with roots approximately at $-2$, $0$, and $2.5$. We must determine the correct polynomial equation from the options

1. The function given is $$g(x) = -x^4 + 2x^3 + 5x^2 - 1$$. 2. To determine the end behavior of the polynomial, we focus on the leading term of highest degree because it dominates

1. The problem is to use the Gauss Elimination method to solve the system of linear equations: $$a x + b y + c z = j$$

1. The user requests to write a solution as it would be done on paper, implying detailed, step-by-step work. Since no specific problem is given, here is a demonstration of solving