🧮 algebra

1. **State the problem:** Find the midpoint of the line segment connecting the points (-2, 3) and (0, 6). 2. **Formula for midpoint:** The midpoint $M$ of a segment with endpoints

1. El problema es simplificar la expresión \( \frac{2}{x} + \frac{3}{x^2} + 1 \). 2. Para sumar fracciones con diferentes denominadores, necesitamos un denominador común. Aquí, los

1. The problem is to write a complex equation and then provide its solution separately. 2. Let's consider the complex equation $$z^2 + (3 - 4i)z + (5 + 2i) = 0$$ where $z$ is a com

1. **Stating the problem:** A jacket is discounted 40%, then an additional 25%, and after a 15% tax, the final price is 103.50. We need to find: a. The original price.

1. **Problem:** Jordan buys sneakers for $28.56 after tax. The store had a 92% discount and 15% tax. 2. **Goal:** Find the original price and the tax paid.

1. **State the problem:** Solve the quartic equation $$x^4 - 5x^2 + 1 = 0$$. 2. **Use substitution:** Let $$y = x^2$$, so the equation becomes $$y^2 - 5y + 1 = 0$$.

1. **State the problem:** Factor the quadratic polynomial $$-x^2 - 22x - 121$$ or determine if it is prime. 2. **Rewrite the polynomial:** To factor more easily, factor out the neg

1. **State the problem:** Factor the polynomial completely: $$-2p^3 + 12p^2 + 54p$$. 2. **Identify the greatest common factor (GCF):** Look at the coefficients and the variable pow

1. **State the problem:** Solve the equation $$25(x + 2)^2 = (x - 7)^2 - 81$$ for $x$. 2. **Rewrite the equation:** Expand both sides.

1. **State the problem:** We are given the function $y = (x+3)(x-1)(x-4)$ and asked to analyze it. 2. **Formula and rules:** This is a cubic polynomial expressed in factored form.

1. **State the problem:** Write as a single logarithm: $$\log_5(x^2) + \log_5(3)$$ 2. **Recall the Product Property of Logarithms:** $$\log_a(MN) = \log_a M + \log_a N$$ which mean

1. **State the problem:** Simplify the expression $\frac{2^5 \times 8}{16} \times 2$. 2. **Recall the rules:** Use the laws of exponents and basic arithmetic operations.

1. The problem is to find the value of $68!$, which is the factorial of 68. 2. The factorial of a positive integer $n$, denoted $n!$, is the product of all positive integers from 1

1. The problem asks to find the value of $200!!$, which is the double factorial of 200. 2. The double factorial of an even number $n$ is defined as the product of all even integers

1. The problem is to evaluate the expression $3^6$. 2. The formula for exponentiation is $a^n = a \times a \times \cdots \times a$ (n times).

1. The problem is to evaluate the cube of 6, which is written as $6^3$. 2. The formula for calculating the cube of a number $a$ is $$a^3 = a \times a \times a$$.

1. The problem is to simplify the expression $X^{12}$. 2. The expression $X^{12}$ means $X$ multiplied by itself 12 times.

1. The problem is to evaluate the expression $3^{10}$. 2. The expression $a^n$ means multiplying the base $a$ by itself $n$ times.

1. **State the problem:** Find the constant of proportionality, complete the table, and write the equation for the proportional relationship given the points:

1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find the constant of proportionality $k$ such that $y = kx$.

1. **State the problem:** We need to sketch the graph of the cubic function $$y = (x-3)(x+1)(x+4)$$. 2. **Formula and important rules:** This is a cubic polynomial in factored form