🧮 algebra

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

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. 2. **Write the formula for tax:** Tax

1. **State the problem:** We have two lines, Line A passing through points (3, 6) and (5, -2), and Line B passing through points (2, 5) and (8, k). We need to find the value of $k$

1. **State the problem:** We need to find the equation of a line perpendicular to line A, which passes through points (2, -5) and (10, -1), and this perpendicular line must pass th

1. The problem gives us two lines: $$y = -5x + 2$$ and $$y = -5x + 6$$. 2. We want to find out if these lines intersect or not.

1. **State the problem:** We have two equations: $$x = 4y + 8$$

1. **State the problem:** A student starts with 50 and buys $x$ reusable water bottles, each costing 6. After buying, the student has 14 left. We need to find $x$. 2. **Set up the

1. **State the problem:** We need to find the current size of the graduating class at Arlington High School, given that it was 100 students a generation ago and is now 26% larger.

1. The problem involves analyzing the behavior of the amount of coffee in the pot over time, represented by $y$ as a function of time $x$. 2. We observe that the graphs start at th

1. **State the problem:** Solve the equation $$8x^2 - 7x + 9 = 2x^2 + 6x + 7$$ for $x$. 2. **Bring all terms to one side:** Subtract $2x^2 + 6x + 7$ from both sides to set the equa

1. **State the problem:** Solve the quadratic equation $8x^2 - 7x + 9 = 2x^2 + 6x + 7$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: