🧮 algebra

1. **State the problem:** We are given the function $f(x) = 3x^2 - 8x$ and want to analyze it. 2. **Identify the function type:** This is a quadratic function of the form $ax^2 + b

1. **State the problem:** Find the natural domain of the function $f(x) = -\frac{16}{x}$. 2. **Recall the domain rules:** The domain of a function is the set of all real numbers $x

1. **State the problem:** Solve the equation $$8x + 4 = 6x + 14$$. 2. **Isolate variable terms:** Subtract $$6x$$ from both sides to get $$8x - 6x + 4 = 14$$.

1. The problem is to analyze the equation without solving for $x$. 2. Typically, to solve for $x$, we isolate $x$ on one side using algebraic operations.

1. **State the problem:** Solve the equation $$27x^3 = (x + 5)^3$$ for $x$. 2. **Recall the formula:** The equation is of the form $a^3 = b^3$, which implies $a = b$ or the cube ro

1. **State the problem:** Alonzo paid a $2 start fee plus $0.12 per minute for a scooter ride, and the total cost was $25.52. We need to find the number of minutes, $x$, he rode. 2

1. **State the problem:** Ivanna has a prepaid card with 20 on it. She buys ribbon at 0.11 per yard and has 16.04 left. We need to find how many yards she bought. 2. **Set up the l

1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = -3 \\ 5x + y = 12 \end{cases}$$

1. **State the problem:** Solve the equation $1 - w = 195$ for $w$. 2. **Recall the additive property of equality:** You can add or subtract the same number from both sides of an e

1. Problem: Vereinfachen und Wurzel ziehen verschiedener Ausdrücke. 1. a) Berechne $\sqrt{32} \cdot \sqrt{8}$.

1. **State the problem:** Solve the equation $$24 = -x + 160$$ for $$x$$. 2. **Use the additive property of equality:** To isolate $$x$$, add $$x$$ to both sides and subtract 24 fr

1. **State the problem:** We need to complete the table for the function $$y = x^2 - 2x - 3$$ by calculating the value of $$y$$ for each given $$x$$ value: $$-4, -3, -2, -1, 0, 1,

1. **State the problem:** Simplify the expression $2\sqrt{3}(2\sqrt{6} + 5)$.\n\n2. **Recall the distributive property:** $a(b + c) = ab + ac$. We will distribute $2\sqrt{3}$ to bo

1. **State the problem:** Find the product and simplify the expression $$-4t^{2}(4t^{2} - t + 1)$$. 2. **Formula and rules:** To multiply a monomial by a polynomial, multiply the m

1. **State the problem:** Find the product and simplify the expression $$(4w + 3)(-3w^2 - 3w - 3).$$ 2. **Recall the distributive property:** To multiply two polynomials, multiply

1. **State the problem:** Evaluate the expression $2r + 3(5k - 6)$ when $x = 2$. 2. **Clarify variables:** The expression contains variables $r$ and $k$, but the problem gives a va

1. Problema este să calculăm produsul a două fracții: $\frac{26}{3} \times \frac{27}{13}$.\n\n2. Formula pentru înmulțirea fracțiilor este: $$\frac{a}{b} \times \frac{c}{d} = \frac

1. **State the problem:** Simplify the expression $-5c - 4c + ab + 10$ given $c=2$ and $d=4$. Note that $d$ is not present in the expression, so it will not affect the simplificati

1. **Problem c)**: Given the proportion $\frac{7}{9} : \frac{4}{7} = \frac{m}{n} : 729$, find $m$ and $n$. 2. **Step 1:** Understand the proportion. The ratio $\frac{7}{9}$ to $\fr

1. **State the problem:** Given the equation $$(a^b)^n = \frac{256}{729}$$, find the values of $a$, $b$, and $n$. 2. **Rewrite the equation using exponent rules:** Recall that $$(a

1. **State the problem:** Solve the equation $$|3x| = 24$$ for $x$. 2. **Recall the absolute value property:** For any real number $a$, $$|a| = b$$ implies $$a = b$$ or $$a = -b$$.