🧮 algebra

1. **State the problem:** Factor the expression $$242f^2 - 32$$ completely. 2. **Identify the type of expression:** This is a difference of two terms, which can be factored using t

1. **State the problem:** Factor the quadratic expression $$27q^2 + 144q + 192$$ completely. 2. **Identify the greatest common factor (GCF):** Look for the largest number that divi

1. **Problem:** Identify the center and radius of the circle given by the equation $$(x - 4)^2 + (y - 1)^2 = 3$$ 2. **Formula:** The standard form of a circle's equation is $$(x -

1. **State the problem:** We have populations of six towns given by exponential functions of the form $P = P_0 (r)^t$, where $P_0$ is the initial population and $r$ is the growth/d

1. **State the problem:** We need to match each exponential function with its corresponding graph based on the function's behavior. 2. **Recall the general form of an exponential f

1. Simplify $2x^5 y \cdot 3x^2 y$. Use the product rule for exponents: $a^m \cdot a^n = a^{m+n}$.

1. **State the problem:** Find the point that is $\frac{1}{3}$ of the way from point A to point B, where A is at $(-3, -2)$ and B is at $(1, 5)$. 2. **Formula used:** The point $P$

1. **State the problem:** Find the equation of a quadratic function in vertex form given the vertex at $(3,-6)$ and points $(1,0)$ and $(5,0)$ on the parabola. 2. **Recall the vert

1. **State the problem:** We need to find the domain and range of the given graph based on the description. 2. **Domain:** The domain is the set of all possible $x$-values for whic

1. **Stating the problem:** A person pays 14.40 for 60 feet of something. We want to find how much extra she will pay for 40 feet.

1. The problem is to solve the proportion $129.5:9 = x:50$ for $x$. 2. Recall that in a proportion $a:b = c:d$, the cross products are equal, so $a \times d = b \times c$.

1. **State the problem:** A car travels 129.5 km using 9 L of gas. We want to find how far it can go with 50 L of gas at the same rate. 2. **Formula used:** Distance traveled is pr

1. **State the problem:** Solve for $a$ in the equation $$\frac{4}{7}a + 1 = 3$$. 2. **Isolate the term with $a$:** Subtract 1 from both sides:

1. **State the problem:** Solve for $k$ in the equation $$\frac{2}{5}k + 4 = 6$$. 2. **Isolate the term with $k$:** Subtract 4 from both sides:

1. The problem is to simplify the expression for $f$ given by: $$f = 15 \times \frac{15}{10} b + 9.24$$

1. **State the problem:** Solve the equation $X - 5 - 6X = 0$ for $X$. 2. **Combine like terms:** Group the $X$ terms together.

1. **State the problem:** Solve the system of linear equations: $$3x + 2y = 12$$

1. Let's clarify the problem: You want to know how a transformation can be written out mathematically. 2. A transformation in math often means applying a function or operation to a

1. **State the problem:** We have a function that takes an input number, subtracts 8 from it, then divides the result by 2. 2. **Write the formula:** If the input is $x$, the outpu

1. **State the problem:** Solve the equation $x + 7 - 2x = 18$ for $x$. 2. **Combine like terms:** The terms involving $x$ are $x$ and $-2x$.

1. The problem asks to write a compound inequality describing the graph shown. 2. The graph shows a number line with a highlighted segment between 0 and 2, including solid dots at