🧮 algebra

1. **State the problem.** Find the limit $\lim_{x\to 3}\frac{x^2-9}{x-3}$.

1. **Problem:** Solve $a^3+a^2=36$ for $a$. 2. **Formula and idea:** First, move everything to one side:

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0$$.

1. The problem is to solve the quadratic equation $2x^2 - 4x - 6 = 0$. 2. The formula to solve quadratic equations is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a

1. **State the problem:** Find the value of $x$ such that $f(x) = 4$ given the function $f(x) = 2x + 2$. 2. **Write the equation:**

1. **State the problem:** We need to find the equation of a line given its graph. 2. **Analyze the graph:** The line crosses the y-axis at $-2$ and the x-axis at $5$.

1. **State the problem:** Solve the system of linear equations: $$2x + y = 8$$

1. **State the problem:** We are given the system of equations: $$2x + 3y = 7$$

1. **State the problem:** We are given two linear equations: $$2x + 3y = 7$$

1. **State the problem:** Evaluate the expression $$6 \div 2(1+2)$$. 2. **Apply the order of operations (PEMDAS/BODMAS):**

1. **Problem:** Solve the system of equations $2x+y=8$ and $2x-y=12$, then find $x$ and $y$. 2. **Formula used:** For a system, we can use elimination by adding the equations to re

1. **State the problem:** Simplify the expression $$\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}$$. 2. **Rewrite division as multiplication by reciprocal:**

1. **State the problem:** Solve the linear equation $16 - 2t = 5t + 9$ for $t$. 2. **Write down the equation:**

1. **State the problem:** Add the two complex numbers $(12+3i)$ and $(14+8i)$. 2. **Formula used:** To add complex numbers, add their real parts and their imaginary parts separatel

1. **Stating the problem:** Simplify the expression $3 - 2\sqrt{5} + 6\sqrt{5}$ and find its equivalent form. 2. **Combine like terms:** The terms $-2\sqrt{5}$ and $6\sqrt{5}$ are

1. **State the problem:** Evaluate the composite functions (g \circ f)(-1), (g \circ f)(0), (f \circ g)(-1), and (f \circ g)(4) using the given graphs of y = f(x) and y = g(x). 2.

1. **State the problem:** Divide the polynomial $$4x^2 - 6x + 8$$ by the binomial $$2x + 3$$ using polynomial long division. 2. **Recall the formula and rules:** Polynomial long di

1. **State the problem:** Simplify the radical expression $\sqrt{24}$. 2. **Recall the rule:** To simplify a square root, factor the number inside into perfect squares and other fa

1. **State the problem:** Simplify the expression $$\sqrt{144m^{16}n^{20}}$$. 2. **Recall the formula and rules:** The square root of a product is the product of the square roots:

1. **State the problem:** Solve for $x$ and $y$ in the equation $$\frac{635}{10} = \left(63 \cdot \frac{x}{100}\right) + \left(65 \cdot \frac{y}{100}\right).$$ 2. **Rewrite the equ

1. The problem states: Simplify the fraction $\frac{635}{10}$ and check if it equals 63. 2. The formula for division of numbers is $\frac{a}{b} = a \div b$.