🧮 algebra

1. **Problem statement:** The perimeter of a rectangle is 16 cm and its area is 12 cm\textsuperscript{2}. Calculate the length and width of the rectangle. 2. **Formulas and rules:*

1. Il problema chiede di semplificare l'espressione $$\left(-\frac{9}{4} x^3 y^4 - \frac{1}{2} x^3 y^4\right) - \left(\frac{5}{8} x^3 y^4 - \frac{5}{4} x^3 y^4\right)$$. 2. Iniziam

1. **State the problem:** Factor the expression $$5y^2z - 42x^4y^5$$ completely. 2. **Identify common factors:** Look for the greatest common factor (GCF) in both terms.

1. **State the problem:** Factor completely the expression $$-46x^3 y z^4 - 29z$$. 2. **Identify the greatest common factor (GCF):** Look at each term:

1. **Problem:** Factor the quadratic expression $x^2 + 7x + 12$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$ where $a=1$, find two numbers that multiply to $c$

1. The problem is to analyze the function $f(x) = x^2 - 7x + 12$. 2. This is a quadratic function of the form $f(x) = ax^2 + bx + c$ where $a=1$, $b=-7$, and $c=12$.

1. **State the problem:** Solve the equation $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$ for $x$. 2. **Write the equation clearly:** $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$

1. **Problem:** Solve the multistep equation $3(2x - 6) + x = 3$ for $x$. 2. **Formula and rules:** Use the distributive property and combine like terms.

1. The problem states the relationship between the number of ravioli $r$ and the number of people $y$ as $y = \frac{1}{4}r$. 2. This means for every 4 ravioli, there is 1 person. T

1. **Problem statement:** A zoo keeper wants to fence a rectangular habitat for goats. The length $L$ should be at least 80 feet, and the perimeter $P$ should be no more than 300 f

1. **State the problem:** We are given inequalities involving $x$ and $y$:

1. Stating the problem: Solve for $x$ in the equation $$\sqrt{x} + 4! + \sqrt{x} = 12$$. 2. Recall that $4!$ (4 factorial) means $4 \times 3 \times 2 \times 1 = 24$.

1. **State the problem:** Solve the inequality $$|x + 6| \leq 8$$ algebraically. 2. **Recall the rule for absolute value inequalities:** For any real number $A$ and positive number

1. **State the problem:** Solve the inequality $$|x - 6| < 7$$ algebraically. 2. **Recall the rule for absolute value inequalities:** For any real number $a$ and positive number $b

1. The problem is to understand and express the number 9.2 × 10^{-11}. 2. This is a number written in scientific notation, where 9.2 is the coefficient and 10^{-11} means we multip

1. The problem states that Maria drives at 50 mph. 2. This is a straightforward statement of speed, so no formula is needed unless we want to calculate distance or time.

1. **State the problem:** Maria and Marc both drive approximately 50 miles to work. Maria drives at 60 mph without stops, while Marc drives at 40 mph but stops for 10 minutes. We n

1. **State the problem:** Elijah and Jonathan each create a model to represent $x$, the number of games their team has left to play after playing 3 of 15 games. 2. **Elijah's model

1. **State the problem:** Find the center and vertices of the ellipse given by the equation $$3x^2 + 2y^2 - 24x + 8y + 20 = 0$$. 2. **Rewrite the equation grouping x and y terms:**

1. **State the problem:** We need to find the result of replacing 8 with 6 in a given expression or context. Since the user only said "6 instead of 8," we assume the problem is to

1. **Problem 1: Analyze the quadratic function** $r(x) = x^2 + 2x - 35$. 2. **Check the vertex location:** The vertex of a parabola $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.