🧮 algebra

1. Let's start by stating the problem: A quadratic equation is a polynomial equation of degree 2, generally written as $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants a

1. **State the problem:** We are given the original ratio of workers to supervisors as 8:2. 2. **Express the original numbers:** Let the number of workers be $8x$ and the number of

1. **State the problem:** The original ratio of students to teachers is 5:1. After 10 more teachers join, the ratio changes to 5:2. We need to find the original number of students.

1. Find the domain of each rational expression. 1.a. Expression: $$\frac{x^2 - 4}{x^2 + 3}$$

1. **State the problem:** Simplify the expression $$6.2 \div \left[ 3.8 + \left\{ 10.6 - \left( 30 - 25 - \frac{4}{5} \times 3.5 + 4.0 \right) \right\} \right]$$. 2. **Calculate th

1. Let's start by understanding the problem: you want a step-by-step explanation for solving a math problem. Since no specific problem was given, I'll demonstrate solving a simple

1. The problem states that the mice population starts at 25,000 and decreases by 20% each year. 2. To model this, we use an exponential decay function: $$P(t) = P_0 (1 - r)^t$$ whe

1. **State the problem:** The population growth of mosquitos follows the uninhabited growth model, which is exponential growth without limits. We are given the initial population $

1. The problem asks for the initial value in the exponential function $$y = 1200 (1 + 0.3)^t$$. 2. The initial value in an exponential function of the form $$y = a(1 + r)^t$$ is th

1. The problem asks to identify which equation does NOT represent exponential decay. 2. Exponential decay occurs when the base of the exponential function $y = a^x$ satisfies $0 <

1. **Find the domain of each rational expression:** **a.** \( \frac{x^2 + 3}{2y} \)

1. The problem asks us to identify which function represents exponential growth from the options given: a. $y=4^x$

1. **State the problem:** Create a polynomial function of degree 4 with an odd leading coefficient and an even constant term.

1. **State the problem:** Simplify the expression \( \frac{3f3mux}{10p2g4} \div \frac{12mvy}{15pd3} \). 2. **Rewrite the division as multiplication by the reciprocal:**

1. **State the problem:** Given the ratio of apples to oranges is 3:5, and if 4 apples are added, the new ratio becomes 1:1. We need to find the value of $x$ and the number of appl

1. The problem states the ratio of apples to oranges is 3 : 5. 2. We represent the number of apples as $3x$ and the number of oranges as $5x$, where $x$ is a positive number repres

1. **State the problem:** Solve the equation $$\frac{4}{x} + 4 + \frac{4x}{x^2 + 4x} = -\frac{5}{x+4}$$ for $x$. 2. **Simplify the terms:** Note that $$x^2 + 4x = x(x+4)$$, so $$\f

1. **State the problem:** Simplify the expression $$\frac{9ab^2}{9ab} \times \frac{5m^2n}{10m^3n^3}$$. 2. **Simplify the first fraction:** $$\frac{9ab^2}{9ab} = \frac{9}{9} \times

1. **State the problem:** Simplify the expression $$\frac{16}{5m^2n} \times \frac{9ab^2}{10m^3n^3}$$. 2. **Rewrite the expression:**

1. **State the problem:** Solve the equation $$\frac{3p}{4} + 3 = \frac{2p}{3} - \frac{1}{6}$$ for $p$. 2. **Eliminate fractions by finding the least common denominator (LCD):** Th

1. **State the problem:** Solve the equation $$3\left(\frac{1}{4}p + 1\right) = \frac{1}{3}\left(2p - \frac{1}{2}\right)$$ for $p$. 2. **Distribute the constants:**