🧮 algebra

1. **State the problem:** Solve the equation $$\sqrt{64x} = x + 12$$ for $x$. 2. **Recall the formula and rules:** The square root function $\sqrt{y}$ returns the non-negative root

1. **State the problem:** Solve the equation $x - 6\sqrt{x} = 16$ for $x$. 2. **Rewrite the equation:** Let $y = \sqrt{x}$. Then $x = y^2$. Substitute into the equation:

1. **State the problem:** We are given that the height of a door is 2.1 feet longer than its width, and the area of the door is 1675.8 square feet. We need to find the width and he

1. **State the problem:** Solve the quadratic equation using the quadratic formula: $$x(2x + 2) = 11$$ 2. **Rewrite the equation in standard form:**

1. **State the problem:** Solve the equation $$\frac{x^2}{5} - x = \frac{1}{5}$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by 5 to clear the denominators:

1. **State the problem:** Simplify the expression $ (2p)^0 $ where $ p > 0 $. 2. **Recall the zero exponent rule:** For any nonzero number or expression $ a $, $ a^0 = 1 $. This me

1. **State the problem:** Solve the quadratic equation $$\frac{3}{5}y^2 + \frac{2}{5}y = \frac{16}{5}$$ for $y$. 2. **Rewrite the equation:** Multiply both sides by 5 to clear deno

1. **State the problem:** Solve the quadratic equation $$\frac{1}{2}x^2 - 5x + 2 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx

1. **State the problem:** Simplify the expression $ (2p)^0 $ where $ p > 0 $. 2. **Recall the zero exponent rule:** For any nonzero number or expression $ a $, $ a^0 = 1 $. This me

1. The problem asks to find the value of $5y^0$ where $y > 0$. 2. Recall the rule: any nonzero number raised to the power of 0 is 1, i.e., $a^0 = 1$ for $a \neq 0$.

1. **State the problem:** Solve the linear equation $85x + 58.5y = 9.7$ for one variable in terms of the other. 2. **Formula and rules:** This is a linear equation in two variables

1. **State the problem:** Evaluate the expression $(-10) + 0$. 2. **Recall the rule:** Adding zero to any number does not change the value of that number. This is called the additi

1. The problem is to evaluate the expression $(-12) + 8$. 2. This is a simple addition problem involving negative and positive integers.

1. The problem is to evaluate the expression $(-18) + 12$. 2. This is a simple addition problem involving integers, where you add a negative number and a positive number.

1. **State the problem:** Simplify the expression $8 \times (4t)^0$. 2. **Recall the rule:** Any nonzero number or expression raised to the power of zero equals 1. That is, for any

1. The problem asks to evaluate the expression $ (m + 2)^0 $ where $ m $ is a positive integer. 2. The key rule here is the zero exponent rule: For any nonzero number $ a $, $ a^0

1. **Problem 2:** A is greater than B by 10%, and B is greater than C by 20%. Given C = 20, find A. 2. **Step 1:** Express the relationships using formulas.

1. **Problem:** Find how long the rocket is in the air given the height function $$h = -16t^2 + 128t$$ where $h$ is height in feet and $t$ is time in seconds. 2. **Formula and rule

1. Stating the problem: A is greater than B by 10%, and B is greater than C by 20%. Given C = 20, find A. 2. Important formulas and rules:

1. **State the problem:** We are given a piecewise linear function $f$ and a table with some missing values. We need to complete the table and explain why the equation $f(x) = 4$ h

1. **Problem statement:** In a company, 67.5% of employees are married. 37.5% of employees are female, and 80% of female employees are married. We need to find the percentage $x$ o