🧮 algebra

1. **State the problem:** Divide the polynomial $$-2x^3 + 5bx^2 - 2b^2x$$ by the binomial $$x - 2b$$. 2. **Recall the formula:** Polynomial division can be done using long division

1. **State the problem:** We are given two numbers, say $x$ and $y$, such that their quotient is $-4$ and their difference is $10$. We need to find the values of $x$ and $y$. 2. **

1. **State the problem:** Simplify the expression $10 + \frac{y}{y}$. 2. **Recall the rule:** For any nonzero $y$, $\frac{y}{y} = 1$. This is because any nonzero number divided by

1. **State the problem:** Solve the equation $4=\frac{x}{5}$ for $x$. 2. **Formula and rule:** To solve for $x$ when it is divided by a number, multiply both sides of the equation

1. The problem is to solve the equation $4 = \frac{x}{ }$ where the denominator is missing. 2. Since the denominator is not provided, we cannot solve for $x$ directly.

1. **State the problem:** Carl bought 19 apples of two varieties: Granny Smith and Gala. The total cost was 5.10. Granny Smith apples cost 0.25 each and Gala apples cost 0.30 each.

1. **Stating the problem:** Solve the system of linear equations number 3: $$\begin{cases} x + 3y = -7 \\ -x + 2y = -8 \end{cases}$$

1. **State the problem:** Solve the equation $40x - 8 = 200$. 2. **Write down the equation:**

1. The problem is to simplify the expression \(\ln y = \ln \sqrt{1 + x^2} + \ln 2\) correctly. 2. Recall the logarithm property: \(\ln a + \ln b = \ln (a \times b)\).

1. **State the problem:** Solve the equation $$ff(x) = gf(2)$$ where $$f(x) = 10 - 3x$$ and $$g(x) = \frac{10}{3 - 2x}$$ with domain restrictions $$x \in \mathbb{R}$$ and $$x \neq

1. **Problem Statement:** Match each equation with its corresponding graph based on the transformations of the square root function. 2. **Recall the base function:** The basic squa

1. **Stating the problem:** We have a function $f$ defined by $f(x) = \frac{x+3}{2x-1}$ with domain $x \in \mathbb{R}, x \neq \frac{1}{2}$. We need to show that $f(f(x)) = x$ and t

1. **State the problem:** Solve the inequality $$-35 < 15 + 10p \geq 25$$. 2. **Understand the compound inequality:** This means two inequalities must hold simultaneously:

1. **State the problem:** We need to find all possible values of $n$ that satisfy the inequality $$-6n < -12$$. 2. **Recall the rule for inequalities:** When dividing or multiplyin

1. **Stating the problem:** Simplify or understand the expression involving the square root function $\sqrt{x}$. 2. **Formula and rules:** The square root function $\sqrt{x}$ is de

1. The problem appears to involve simplifying or solving an expression involving $y + 3$, $1$, and $y - 2$. However, the exact operation is unclear from the input. 2. Assuming the

1. **State the problem:** Simplify the expression $$\sqrt{\left(6-\frac{3}{2}\right)2}+\sqrt{-t}$$. 2. **Simplify inside the first square root:** Calculate $$6-\frac{3}{2}$$.

1. **Stating the problem:** We want to factor a quadratic polynomial of the form $ax^2 + bx + c$ using the quadratic formula. 2. **Formula used:** The quadratic formula to find roo

1. Let's solve a random algebra problem: Solve for $x$ in the equation $$2x + 3 = 11$$. 2. The formula to isolate $x$ is to subtract 3 from both sides and then divide both sides by

1. **State the problem:** We are given two matrices: Matrix A (3x3):

1. The problem involves identifying the equation of a straight line passing through points (-6, 6) and (6, -6). 2. The formula for the slope $m$ of a line passing through points $(