🧮 algebra

1. Calculate the following expressions involving square roots. 2. Use the property \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) to simplify each fraction.

1. **Problem statement:** Find the value of $k$ such that the area of the triangle with vertices $(2, -6)$, $(5, 4)$, and $(k, 4)$ is 35 square units. 2. **Formula for area of tria

1. **State the problem:** We are given a triangle with vertices at points $(2, -6)$, $(5, 4)$, and $(k, 4)$, and the area of this triangle is 35 square units. We need to find the v

1. Masalah ini meminta kita untuk menghitung solusi setelah substitusi dilakukan. 2. Langkah pertama adalah menuliskan persamaan yang sudah disubstitusikan.

1. The problem is to solve the equation or expression given by the user. However, no specific equation or expression was provided. 2. To solve an equation, we typically isolate the

1. **State the problem:** Express 0.000407 correct to two significant figures. 2. **Understand significant figures:** Significant figures are digits that carry meaning contributing

1. **State the problem:** Express 0.0000407 correct to two significant figures. 2. **Understand significant figures:** Significant figures are the digits in a number that carry mea

1. **Ratio**: A ratio compares two quantities showing how many times one value contains or is contained within the other. It is written as $a:b$ or $\frac{a}{b}$. For example, if t

1. **Problem:** Find the limit of $$\frac{6x^2 - 17x + 12}{2x^2 - x - 3}$$ as $$x \to 4$$. 2. **Formula:** For polynomial limits at a point, substitute the value directly if the de

1. The problem asks to find the value of the expression $abcd$. 2. To solve this, we need the values of $a$, $b$, $c$, and $d$.

1a. Rationalize the following expressions: 1. $$\frac{2}{3-\sqrt{2}} + \frac{1}{3+\sqrt{2}}$$

1. Problem: Given $f(x)=-7x^2-5$ and $g(x)=2x-1$, solve $f(g(x))=-68$. 1. Formula and rule: $f(g(x))$ means substitute $g(x)$ into $f$, so compute $f(g(x))$.

1. **State the problem:** We need to find a number such that when it is multiplied by 3 and then increased by 11, the result is 35. 2. **Set up the equation:** Let the number be $x

1. Problem (ii): One complex solution to $f(x)=x^2+2x+5$ is $w_1=-1+2i$. Determine the other solution $w_2$ and state the relationship between $w_1$ and $w_2$. 2. Formula and rule:

1. Problem (ii): One complex solution to $f(x)=x^2+2x+5$ is $w_1=-1+2i$. Determine the other solution $w_2$ and state the relationship between $w_1$ and $w_2$. 2. Formula and rules

1. **Problem Statement:** Find the answers for the empty boxes in the given problem (assuming the problem is related to the previous date calculation). 2. **Step 1: Identify the mi

1. Problem 1: Determine the other solution $w_2$ of $f(x)=x^2+2x+5$ given $w_1 = -1 + 2i$, and state the relationship between $w_1$ and $w_2$. 2. Formula and rule: For a polynomial

1. **Problem:** Given one complex solution to the quadratic equation $f(x) = x^2 + 2x + 5$ is $w_1 = -1 + 2i$, find the other solution $w_2$ and state the relationship between $w_1

1. **State the problem:** Solve the equation $$(x - 2500) \times 40\% - 21x = 900$$ for $x$. 2. **Rewrite the percentage as a decimal:** $40\% = 0.40$.

1. **State the problem:** Solve the equation $-9 + 4q = -9q + 4$ for $q$. 2. **Write down the equation:**

1. **State the problem:** Solve for the variable $w$ in the equation $$4 \left(\frac{-6w - 9}{27} \right) - 2 = 2.$$\n\n2. **Rewrite the equation:** The equation is $$4 \times \fra