🧮 algebra

1. **State the problem:** Rationalize the denominator of the expression $$\frac{\sqrt{g}}{\sqrt{g} + \sqrt{k}}$$ where $g$ and $k$ are positive real numbers. 2. **Recall the formul

1. **State the problem:** Simplify the expression $$(5 + 2\sqrt{2})(3\sqrt{2} - 3)$$. 2. **Use the distributive property (FOIL) to expand:**

1. **State the problem:** Simplify the expression $14\sqrt{23} \cdot 5\sqrt{2}$. 2. **Recall the multiplication rule for radicals:** For any positive numbers $a$, $b$, $c$, and $d$

1. **State the problem:** Calculate the product of $3$ and the absolute value of $\frac{7}{2}$. 2. **Recall the absolute value rule:** The absolute value of a number is its distanc

1. **Stating the problem:** Solve the equation $$5^2 - x = \frac{1}{25}$$ for $$x$$. 2. **Recall the formula and rules:** We know $$5^2 = 25$$.

1. Problema 29: Fracțiile scrise de Andrei, Bianca și Corina. - Andrei scrie fracții de forma $\frac{a}{8}$ cu $a \mid 8$. Divizorii lui 8 sunt $1, 2, 4, 8$.

1. **State the problem:** We want to find how many children bought tickets to the fair given that adult tickets cost $25, children's tickets cost $10, 1000 people attended, and tot

1. Solve the system: $$-x = 4 - y$$

1. **State the problem:** Krystal has some $1 coins and $2 notes. The number of $2 notes is 4 times the number of $1 coins. The total value of all coins and notes is $54. We need t

1. **State the problem:** Sumita and Carrie start with ribbons of the same length $x$ cm. 2. Sumita used $75.6$ cm, so she has $x - 75.6$ cm left.

1. **State the problem:** We have a mixture of gold and silver in the ratio $3:7$ by weight, and the total weight is $250g$. We need to find the weight of gold. 2. **Formula and ru

1. **Problem statement:** Solve the absolute value inequalities and express solutions in interval notation. ---

1. **Problem statement:** Solve the absolute value inequalities and express the solutions in interval notation. ---

1. **State the problem:** Reflect point $A(2,1)$ about the line $y = -x + 1$ and find the coordinates of the reflected point $A'$. 2. **Identify the line equation in standard form:

1. **Problem:** Given matrices $A = \begin{pmatrix}4 & 2 \\ 5 & 3\end{pmatrix}$ and $B = \begin{pmatrix}-5 & -2 \\ 4 & 1\end{pmatrix}$, find: a) $A^2$

1. **State the problem:** We want to factor the expression $x^2$. 2. **Recall the factoring rules:**

**Factoring $x^2$** **1. Understanding the Problem**

1. **Problem Statement:** Find the following for matrices $A$ and $B$:

1. **State the problem:** Solve the quadratic equation $x^2 + 5x + 6 = 0$. 2. **Formula and rules:** The quadratic equation is generally solved using the quadratic formula:

1. Let's clarify what it means to "cancel out zeros" in math. 2. Zeros in multiplication can make the entire product zero, but you cannot cancel zeros like you cancel common factor

1. The problem shows two ratios or fractions: $\frac{3}{5}$ and $\frac{3}{6}$, and numbers $721$ and $7$ are also given. 2. We need to understand the relationship or compare these