🧮 algebra

1. **State the problem:** Solve the equation $$2x(x - 1) = (3x - 1)(x + 1) + 4$$ for $x$. 2. **Expand both sides:**

1. Write the equation of each line in slope-intercept form $y=mx+b$ given a point $(x_1,y_1)$ and slope $m$. 2. For problem 3: Point $(1,3)$, slope $m=-5$.

1. Write the equation of the line passing through (5, -8) with slope $m = -3$ in slope-intercept form $y = mx + b$. 2. Use the point-slope form $y - y_1 = m(x - x_1)$ with point $(

1. **Solve for y in the equation:** $$y - 3 = - \frac{2}{5} (10x - 5)$$

1. Stating the problem: Simplify the expression $$\frac{10}{6+2x} + \frac{3}{x+3}$$. 2. Factor the denominator of the first fraction: $$6+2x = 2(3+x)$$.

1. The problem is to simplify the expression $$\frac{10}{6} + 2x + \frac{3}{x} + 3$$. 2. First, simplify the fraction $$\frac{10}{6}$$ by dividing numerator and denominator by 2:

1. **Graph the function** $-5x + 2y = -10$. Step 1: Rewrite in slope-intercept form $y = mx + b$.

1. **State the problem:** Simplify the expression $$\frac{10}{6} + 2x + \frac{3}{x} + 3$$ completely. 2. **Simplify the fraction $$\frac{10}{6}$$:**

1. The problem is to solve the equation $$\sqrt{1-x} \sqrt{x} + \frac{2}{3} = -\frac{2}{3}$$. 2. First, isolate the square root term by subtracting \(\frac{2}{3}\) from both sides:

1. The problem is to solve the equation $x^2 - 5x + 6 = 0$. 2. Start by factoring the quadratic expression. We look for two numbers that multiply to $6$ and add to $-5$.

1. The problem is to analyze the polynomial $$x^3 + ax^2 + bx + 1$$ where $a$ and $b$ are constants. 2. We can start by stating the polynomial explicitly: $$P(x) = x^3 + ax^2 + bx

1. **State the problem:** Simplify and understand the logarithmic expressions and their derivatives using the laws of logarithms. 2. **Recall the laws of logarithms:**

1. The problem is to understand the letter 'a' as it stands alone without any mathematical context. 2. Since 'a' is a single variable or symbol, there is no calculation or equation

1. Let's clarify the question: "Why did you put 5 up from 6?" This likely refers to an expression or fraction where 5 is placed above 6, such as $\frac{5}{6}$. 2. In mathematics, w

1. The problem is to solve the equation $$2x^2 - 4x - 6 = 0$$ for $x$. 2. First, simplify the equation by dividing all terms by 2 to make the coefficients smaller:

1. **State the problem:** Simplify the expression $$\frac{-\frac{1}{3} a^3 b^5}{\frac{6}{5} a^2 b^2}$$. 2. **Rewrite the division as multiplication by the reciprocal:**

1. The problem is to solve an equation or expression using a method, but the specific problem is not provided. 2. To demonstrate a method, let's consider solving a simple linear eq

1. The problem is to analyze the equation $y = (x + y + 1)^2$ and understand its behavior. 2. Start by expanding the right side: $$y = (x + y + 1)^2 = (x + y + 1)(x + y + 1) = x^2

1. The problem is to multiply $2x$ by $x$. 2. Recall that when multiplying terms with the same base, you add the exponents: $x^1 \times x^1 = x^{1+1} = x^2$.

1. **State the problem:** We are given the equation $$x^2 \sqrt{x^2 - c^2} = c^2 \sqrt{x^2 - c^2} + 39$$ where $c$ is a positive constant. We need to find one of the solutions for

1. **State the problem:** Factorize fully the expression $$15y^4 + 20xy^3$$. 2. **Identify the greatest common factor (GCF):**