🧮 algebra

1. **State the problem:** Rearrange each linear equation into the form $y = mx + c$ where $m$ is the slope and $c$ is the y-intercept. 2. **Formula and rules:** The slope-intercept

1. **State the problem:** Jack starts with 450 in his savings account and withdraws 25 each week. He wants to have at least 100 left at the end of the summer. We need to find how m

1. Planteamos el problema: hallar todos los valores $c$ en el intervalo $[0,2]$ tales que $f(c)=0$. 2. Para resolver $f(c)=0$, necesitamos conocer la expresión explícita de $f(x)$,

1. **State the problem:** Jack starts with 450 in his savings account and wants to have at least 100 left after withdrawing 225 each week. 2. **Set up the inequality:** Let $w$ be

1. The problem is to simplify $\sqrt{\sqrt{625}}$. 2. Recall that $\sqrt{a}$ means the square root of $a$, and $\sqrt{\sqrt{a}}$ means the square root of the square root of $a$.

1. **State the problem:** Calculate the fifth root of the product of -32 and -243, i.e., find $$\sqrt[5]{(-32) \times (-243)}$$. 2. **Recall the formula:** The nth root of a produc

1. **State the problem:** Find the roots of the cubic polynomial $$x^3 + 4x^2 - 17x - 60$$ and verify them. 2. **Formula and rules:** To find roots of a cubic polynomial, we can tr

1. **Problem statement:** Given the cubic equation $x^3 - 8x - 1 = 0$, it has a real root between two consecutive integers $n$ and $n+1$. The function values $f(x) = x^3 - 8x - 1$

1. **State the problem:** Solve the equation $$\sqrt{n} + 10 + 2 + \sqrt{n} - 5 = 0$$ for $n$. 2. **Simplify the equation:** Combine like terms.

1. **State the problem:** Solve the equation $$1 - \left(x + \frac{x}{2} + \frac{x}{5}\right) = 20$$ for $x$. 2. **Rewrite the equation:** Combine the terms inside the parentheses:

1. **State the problem:** Simplify the algebraic expression $$\frac{3x^2}{x+4} + \frac{5z}{x+64}$$ and $$\frac{2x-3}{x^2+5x+6} + \frac{4}{x+3}$$ and $$\frac{6}{x+4} + \frac{2x+8}{x

1. **State the problem:** Simplify the function $$f(x) = \frac{1 - \sin^2(x)}{\cos(x)}$$. 2. **Recall the Pythagorean identity:** $$\sin^2(x) + \cos^2(x) = 1$$, which implies $$1 -

1. **State the problem:** We need to find the rule for the table where $X$ is the input and $Y$ is the output: $$\begin{array}{c|c}

1. **State the problem:** Graph the two linear equations on the same coordinate plane: a) $2x + y = 2$

1. **State the problem:** We are given the equation $y = 2$ and asked to analyze its graph and points related to it. 2. **Rewrite in standard form:** The equation $y = 2$ can be wr

1. **State the problem:** Find the slope of the line passing through the points $(-4, 0)$ and $(4, -2)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1, y_1

1. **State the problem:** Solve the equation $$3 \cdot 4^x + 6 \cdot 2^x = 24$$ for $x$. 2. **Recall the formula and rules:** Note that $4^x = (2^2)^x = 2^{2x}$. This allows us to

1. **State the problem:** Find the slope of the line passing through the points (0, 4), (1, 2), and (-1, 6). 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$

1. **State the problem:** We need to determine which pizza is the better buy by comparing the unit price per square inch for each pizza. 2. **Formula for area of a circle:**

1. **State the problem:** Solve the equation $$4 \log(x) = 2 \log(x) + \log(4) + 2$$ for $x$. 2. **Recall logarithm properties:**

1. **State the problem:** Solve the equation $$\log_3(x + 2) + \log_3(x - 1) = 2$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms with the same base can be co