🧮 algebra

1. **Stating the problem:** We want to create a fairer speeding fine system combining elements from New York and Massachusetts systems.

1. **Problem Statement:** We want to compare the New York and Massachusetts speeding fine systems based on miles per hour (mph) over the speed limit.

1. **State the problem:** We need to find which values of $x$ from the given list make the function $f(x) = x^2 - 5x - 36$ equal to zero. 2. **Formula and rule:** A zero of the fun

1. **State the problem:** Simplify the expression $10c + c^2 - 3c^3$. 2. **Identify terms:** The expression contains three terms: $10c$, $c^2$, and $-3c^3$.

1. **State the problem:** Simplify the expression $$\frac{10c + c^2 - 3c^3}{5c^2 - 6c - 8}$$. 2. **Rewrite the expression:** Arrange terms in descending powers of $c$ for clarity:

1. **State the problem:** We need to find an expression for the function $$y = 6x^3 - \frac{4}{x^2}$$. 2. **Understand the components:** The function consists of two terms: a polyn

1. **State the problem:** We need to find an expression for the area of a rectangular field. 2. **Given:** The length of the field is $4x + 2$ meters.

1. **Work out the multiplications and simplify each:** **a.** $\frac{3}{5} \times \frac{6}{7}$

1. Problem: Simplify the expression \((15^{\frac{3}{8}})(81^{\frac{1}{34}})(15^{\frac{4}{3}})(81^{\frac{1}{7}})(15^{\frac{7}{8}})(81^{\frac{1}{7}})\). 2. Use the property of expone

1. **State the problem:** Simplify the expression $$\frac{1}{\sqrt{5} - 2} \times \frac{1}{\sqrt{5} + 2}$$. 2. **Recall the formula:** When multiplying expressions with conjugates

1. **Problem:** Simplify the expression $$ (15^{\frac{3}{8}})(81^{\frac{1}{34}})(15^{\frac{3}{4}})(81^{\frac{1}{7}})(15^{\frac{5}{8}})(81^{\frac{1}{7}}) $$ 2. **Formula and rules:*

1. **State the problem:** Simplify the expression $$\frac{36 \pi}{5} \times \frac{8}{20} = \frac{1}{\pi} - \frac{1}{\pi} \times \frac{5}{8}.$$\n\n2. **Simplify the left side:** Mul

1. Let's create a real-life situation involving algebra. 2. Problem: Suppose you are buying apples and oranges. Apples cost $2 each and oranges cost $3 each. You buy a total of 10

1. **State the problem:** Solve the equation $2x + 5 = 9$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation. We do this by per

1. **State the problem:** Given the system of equations $$2x + y = \frac{1}{\sqrt{5 - 2}}, \quad 2x - y = \frac{1}{\sqrt{5 + 2}}$$

1. We are asked to find the composition \((g \circ f)(x)\), which means we substitute \(f(x)\) into \(g(x)\). 2. Given functions:

1. **Stating the problem:** Evaluate the expression $7\log 4 \cdot 2\log 5 + 7\log \frac{49}{25}$.

1. **Stating the problem:** We want to express $\log_6 18$ in terms of $x = \log_6 6$ and $y = \log_2 2$. 2. **Recall the given values:**

1. 문제: 식 $$(x+y)^2 - 9(x+y)(2x - y) + 20(2x - y)^2$$ 을 인수분해하시오. 2. 치환: $x + y = A$, $2x - y = B$ 로 치환하면 식은 $$A^2 - 9AB + 20B^2$$ 가 됩니다. (㉠, ㉡)

1. **State the problem:** Simplify the expression $$\frac{X^{-4} \cdot X^{3}}{X}$$. 2. **Recall the laws of exponents:**

1. 문제: 다항식 $x^2 - y^2 + x + 7y - 12$가 $A(x + y - 3)$의 형태로 표현될 때, 다항식 $A$를 구하시오. 2. 주어진 식을 인수분해하는 방법을 사용합니다. 우선 $x^2 - y^2$는 차의 제곱 꼴로 인수분해할 수 있습니다: