🧮 algebra

1. **State the problem:** Simplify the expression $2a + 5b + 4a - 3b$. 2. **Group like terms:** Combine the terms with $a$ and the terms with $b$ separately.

1. The problem is to understand the expression $2x + 3$. 2. This is a linear algebraic expression where $2x$ means 2 times the variable $x$.

1. The problem is to factor the quadratic expression $6x^2 + 11x + 4$. 2. First, identify coefficients: $a=6$, $b=11$, and $c=4$.

1. El problema es completar la factorización de un polinomio cuadrático usando la ecuación de segundo grado. 2. La ecuación general de segundo grado es $$ax^2 + bx + c = 0$$.

1. The problem is to simplify the expression $5x + 18x$. 2. Both terms have the variable $x$, so we can combine them by adding their coefficients.

1. **Planteamiento del problema:** Analizar la función $$f(x) = \frac{x^2 - 9}{4 - x^2}$$ y describir sus características principales, incluyendo asíntotas, interceptos y comportam

1. The problem is to solve the logarithmic equation $\log_x \frac{1}{16} = -\frac{1}{2}$.\n\n2. Recall that $\log_a b = c$ means $a^c = b$. Applying this, we get:\n$$x^{-\frac{1}{2

1. Statement of the problem. Suppose a sample of a substance decayed to 77.8% of its original amount after 300 days, and we are asked to find the half-life and the time to reach on

1. **Problem:** If the roots of the equation $ax^2 + cx + c = 0$ are in the ratio $p : q$, show that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0.$$ **Solutio

1. The problem is to factor the expression $x^2$. 2. Notice that $x^2$ is a perfect square, which means it can be written as $x \times x$.

1. **State the problem:** We have a circle given by the equation $$x^2 + y^2 - 10x + 8y + 25 = 0$$ and a line given by $$y = kx$$. We want to find the values of $k$ for which the l

1. **State the problem:** Solve the system of linear equations: $$1.2y + 10.4 = -8.2x$$

1. **State the problem:** Solve the system of equations: $$3x - 4y = 23 \quad (1)$$

1. **Problem statement:** (a) Given a rectangular garden with area 80 m², width $y$ metres, length $x$ metres, and a path of uniform width 2 metres around it.

1. Bentuk sederhana dari $\left(6^{-2} a\right)^3 : \left(12^2 a^3\right)^{-2}$ adalah... Langkah-langkah:

1. The problem is to simplify the expression $2\ln(e^3) + \ln(e^2)$.\n\n2. Recall the logarithm power rule: $\ln(a^b) = b\ln(a)$. Since $\ln(e) = 1$, we can simplify each term:\n\n

1. Bentuk sederhana dari $\frac{6^{-2} a^3}{(12^1 a^3)^{-2}}$ adalah... Langkah-langkah:

1. The problem is to analyze the function $f(n) = \frac{n}{20 - 4n}$.\n\n2. First, identify the domain. The denominator cannot be zero, so solve $20 - 4n = 0$ which gives $n = 5$.

1. **State the problem:** Solve the equation $$\frac{4r-3}{6r+1} = \frac{2r-1}{3r+4}$$ for $r$. 2. **Cross-multiply to eliminate the fractions:**

1. The problem is to find the domain of a function, but the function is not specified. 2. The domain of a function is the set of all possible input values (x-values) for which the

1. The problem is to find the domain of the function $$f(n) = \frac{n}{20} - 4n$$. 2. The function is a rational expression where the numerator is $$n$$ and the denominator is $$20