🧮 algebra

1. The problem asks us to identify the equation of a polynomial $p(x)$ based on the shape and roots of its graph. 2. The polynomial's roots are at $x = 0$, $x = 3$, and $x = -\frac

1. Stating the problem: Simplify or factor the quadratic expression $x^2 + 4x + 3$. 2. To factor, we look for two numbers whose product is $3$ (the constant term) and whose sum is

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