🧮 algebra

1. **Statement of the problem:** Calculate the following expressions:

1. **State the problem:** We have the formula for sag $S$ of a wire stretched between two points:

1. **State the problem:** We are given the sag formula for a wire stretched between two points: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$

1. **Stating the problem:** We are given the formula for sag $S$ in terms of wire length $L$ and distance $d$: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$

1. The problem asks for the interpretation of the x-intercept of the graph showing the diver's height above water over time. 2. The x-intercept of a graph is the point where the fu

1. **State the problem:** We need to find the equation of a line $r$ in the $xy$-plane that has a slope of 4 and passes through the point $(0,6)$. 2. **Formula used:** The slope-in

1. The problem is to simplify the expression $\log_p(8p^5)$. 2. Recall the logarithm product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$.

1. **Problem:** Solve for $x$ in the equation $$8^{2x+1} = \frac{128^{x-2}}{4^x \times 16}$$ 2. **Step 1: Express all terms with base 2**

1. Problem: Calculate the cube root of 360, denoted as $\sqrt[3]{360}$.\n\n2. Formula and rules: The cube root of a number $x$ is a number $y$ such that $y^3 = x$. We can simplify

1. **State the problem:** (a) Solve for $r$ in the equation $$4 \pi r^3 = 45$$

1. **State the problem:** Simplify the expression $$\frac{140+35x}{150+30x}$$ and check if it can be written as $$\frac{14+7x}{15+6x}$$. 2. **Identify common factors:** Look for co

1. **State the problem:** Solve the equation $$\frac{140+35x}{150+30x} = \frac{28}{27}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cros

1. **Stating the problem:** We are given the cubic polynomial $x^3 - 2x^2 + 2kx + 8$ and need to analyze or solve it depending on the context (e.g., find roots, factor, or determin

1. **State the problem:** We are given the function $f(x) = \frac{3}{4}x + 10$ and want to understand its properties. 2. **Formula and explanation:** This is a linear function of t

1. The problem is to understand and simplify the function $f(x) = \frac{2}{4}x + 10$. 2. The formula given is a linear function in the form $f(x) = mx + b$, where $m$ is the slope

1. **State the problem:** A notebook costs four times more than a pencil. The pencil costs 30 cedis less than the notebook. Find the price of the pencil. 2. **Define variables:** L

1. **State the problem:** Solve the equation $$\frac{1}{2} x^2 = 128$$ for $x$. 2. **Formula and rules:** This is a quadratic equation in the form $$ax^2 = b$$ where $a = \frac{1}{

1. **State the problem:** We need to find the quotient of the functions $f(x) = x^3 + x^2 + 2$ and $g(x) = x - 1$, i.e., compute $\frac{f(x)}{g(x)}$. 2. **Formula and method:** To

1. **Problem:** Given $A \times B = 24$, $C \times D = 32$, $B \times D = 48$, and $B \times C = 24$, find $A \times B \times C \times D$. 2. **Formula and approach:** We want to f

1. The problem is to expand and simplify the expression $ (x + 2)(x + 3)(x + 6) $. 2. We start by multiplying the first two binomials: $ (x + 2)(x + 3) $. Using the distributive pr

1. **State the problem:** Find the equation of line H in the form $y = mx + c$ given two points on the line: $(-10, 70)$ and $(10, -30)$. 2. **Formula used:** The slope $m$ of a li