🧮 algebra

1. **State the problem:** We have two functions $P(x)$ and $Q(x)$, both with degree 7, and their end behaviors are given: - $P(x) \to -\infty$ as $x \to +\infty$ and $P(x) \to +\in

1. **State the problem:** Solve the quadratic equation $$3x^2 = -8x - 2$$. 2. **Rewrite the equation in standard form:** Move all terms to one side:

1. **State the problem:** Gayle wants to buy a bicycle costing 119.90. She has saved 0.7 of the total cost. We need to find how much she has saved so far. 2. **Formula used:** To f

1. El problema es encontrar el vértice de la función cuadrática $$f(x) = x^2 - 2x + 3$$ y graficarla. 2. La fórmula para encontrar la coordenada $x$ del vértice de una parábola dad

1. **State the problem:** Solve the equation $$\frac{1}{6}(x + 6) = 11$$ for $x$. 2. **Use the distributive property:** Multiply both sides by 6 to eliminate the fraction.

1. **State the problem:** Solve the equation $$\frac{1}{6}(x + 6) = 11$$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation. Si

1. **Problem 1:** Find $x$ when $y = -x + 3$ and $y = 2$. 2. Since both expressions equal $y$, set them equal to each other:

1. The problem states that Angelia's savings account amount is given by the expression $$148.02(1.04)^{t-10}$$ where $t$ is the number of years since the account was opened. 2. We

1. **State the problem:** We want to find an equation that shows buying one item at regular price $x$ and a second identical item at 50% off is the same as buying both items for 25

1. The problem is to simplify the given expression or answer. 2. To simplify an expression, we combine like terms, factor where possible, and reduce fractions.

1. **State the problem:** Solve the equation $$\sqrt{5x + 3} = \sqrt{2x + 4} + 5$$. 2. **Isolate the square root terms:** We want to isolate one square root to eliminate them by sq

1. **State the problem:** Given the equation $x + 9 = 3y^2$, determine the functional relationship between $x$ and $y$. 2. **Rewrite the equation:**

1. **Problem:** Find the real or non-real solutions to the equation $$x^2 + 0.5x - 14 = 0$$. 2. **Formula:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ wh

1. **State the problem:** We are given the function $f(x) = \frac{2}{x+1}$ and we want to understand its behavior. 2. **Formula and rules:** This is a rational function where the n

1. Problema: Calcula la raíz cúbica de 243, es decir, encuentra $\sqrt[3]{243}$. 2. Fórmula: La raíz cúbica de un número $a$ es un número $x$ tal que $x^3 = a$.

1. **Stating the problem:** We need to determine which rooms can hold the planned number of guests without exceeding the hotel's limit. 2. **Understanding the problem:** Each room

1. State the problem. This month's commodity index decreased by one-quarter of last month's index to 60.

1. **State the problem:** This month’s commodity index is 60, which is one-quarter less than last month’s index. We need to find last month’s index. 2. **Set up the equation:** Let

1. **State the problem:** We want to factor the expression $$3^{2x} + 3^x - 6$$ into the form $$(3^x + a)(3^x + b)$$ and find the values of $a$ and $b$. 2. **Rewrite the expression

1. **Problem statement:** We want to find the labeled price $x$ of a dress such that after giving a 20% discount, the seller still makes a 25% profit on the cost price of 80. 2. **

1. **State the problem:** Elizabeth's subtotal is $x$ dollars. A tax of 6% is added, and then a tip of 21% is applied to the total after tax. We need to find an expression for the