🧮 algebra

1. **State the problem:** We are given the family of second order surfaces (SOS) defined by the equation $$3x^2 + y^2 + 2z^2 + 6xy + 2xz + 4yz + ky + z + 1 = 0$$ and we want to dis

1. **State the problem:** Solve the equation $$\frac{(x-3)(x+1)}{\sqrt{\ln(x-1)}} = 0$$ and find the arithmetic average of its roots. 2. **Understand the equation:** A fraction equ

1. **Problem Statement:** Find the domain and range of the function $$f(x) = \sqrt{7x^2 + 25} + 9$$. 2. **Understanding the function:** The function involves a square root, so the

1. **State the problem:** Find the domain and range of the function $$f(x) = \sqrt{7x^2 + 25} + 9$$. 2. **Domain:** The domain of a function involving a square root requires the ex

1. **Stating the problem:** We are given complex numbers $z_1 = -53i$ and $z_2 = 4 - 3i$. We need to find:

1. **State the problem:** Solve the inequality $$\ln(x-1) + \ln(x-3) > \ln(3)$$ for $x$. 2. **Use logarithm properties:** Recall that $$\ln(a) + \ln(b) = \ln(ab)$$ for $a,b>0$.

1. **State the problem:** Calculate the value of the expression $2\sqrt{0.25} + \sqrt{3}$. 2. **Recall the square root properties:**

1. **State the problem:** Simplify the expression $2\sqrt{x}0.25 + \sqrt{x}3$. 2. **Rewrite the expression:** The expression can be written as $2 \times \sqrt{x} \times 0.25 + \sqr

1. The problem is to solve for $x$ in the equation $ax + b = 0$ where $a$ and $b$ are constants. 2. The formula to solve a linear equation of the form $ax + b = 0$ is:

1. **State the problem:** Simplify the expression $$(2q + \frac{1}{2})(3p + 5q)$$. 2. **Formula used:** Use the distributive property (FOIL method) for multiplying two binomials: $

1. **State the problem:** Simplify the expression $2q + \frac{1}{2}$. 2. **Understand the expression:** The expression consists of two terms: $2q$ and $\frac{1}{2}$. These terms ca

1. Let's start by clarifying the problem you want to solve. Since you didn't specify, I'll assume you want a general step-by-step solution for a common algebra problem, such as sol

1. **Problem:** Find the solution set of the equation $$|3 + 2x| + |2x + 3| = 0$$. 2. **Formula and rules:** The absolute value function $$|a|$$ is always non-negative, i.e., $$|a|

1. The problem asks to rewrite the expression $9 \cdot 9^2$ in the form $9^n$. 2. Recall the exponent property for multiplication with the same base: $$a^m \cdot a^n = a^{m+n}$$ wh

1. **State the problem:** Solve for $x$ in the equation $$8^7 = 8^3 \cdot 8^x$$. 2. **Recall the exponent rule:** When multiplying powers with the same base, add the exponents: $$a

1. **State the problem:** Determine the sign of the expression $$\frac{x}{y} \cdot 7y^3$$ given that $$x < 0$$ and $$y > 0$$. 2. **Rewrite the expression:**

1. The problem asks for the sign of the expression $3^0 \cdot 0^{26}$. 2. Recall the rules for exponents:

1. The problem asks for the sign of the expression $$-3.2b \cdot \left(\frac{c}{b} \cdot 0\right)$$. 2. Important rule: Any number multiplied by zero is zero.

1. The problem is to express a system or equation in matrix form for clarity and better visualization. 2. Matrix form typically involves writing the system as $AX = B$, where $A$ i

1. **Problem:** Solve the equation given by the determinant for part (a): $$\begin{vmatrix} 6 & 2 \\ 3 & x \end{vmatrix} = 0$$

1. **State the problem:** We are given that the product of two numbers is $-\frac{28}{27}$ and one of the numbers is $-\frac{4}{9}$. We need to find the other number. 2. **Formula