🧮 algebra

1. **State the problem:** Given $x = 3 + \sqrt{8}$, find the value of $$x^4 + \frac{a}{x^4}$$. However, the problem does not specify the value of $a$. Assuming $a=1$ for the expres

1. **State the problem:** We want to find the values of $a$ for which both roots of the quadratic function $f(x) = x^2 + ax + 1$ are greater than 0. 2. **Recall the quadratic formu

1. **State the problem:** Solve the system of equations for $x$ and $y$: $$2x + y = 8$$

1. **State the problem:** There are two candidates in an election who received 51% and 26% of the total electorate's votes respectively. Additionally, 4600 voters did not vote. We

1. **State the problem:** Solve the equation $$\frac{15}{x-1} = \frac{19}{x+4}$$ for $x$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross prod

1. **Problem Statement:** Construct a forward difference table for the given values of $x$ and $y$, then find $\Delta^2 f(5)$ and $\Delta^3 f(10)$. 2. **Given Data:**

1. **State the problem:** Solve the equation $$\binom{n}{15} \frac{19}{x-1} = \frac{19}{x+4}$$ for $x$. 2. **Understand the equation:** The binomial coefficient $\binom{n}{15}$ is

1. **State the problem:** We want to find values of $a$ such that the quadratic equation $$x^2 - (a^2 - 2)x - a^2 + 3a + 2 = 0$$ has one root greater than 1 and the other root less

1. The problem is to solve the equation $6 = 2m - n$ for $m$. 2. We want to isolate $m$ on one side of the equation.

1. The problem is to verify if the algebraic rearrangements and formulas given for various expressions involving variables $m$, $n$, $p$, $x$, and $y$ are correct. 2. The general r

1. Énonçons le problème : Étudier les prix signifie généralement analyser une fonction qui modélise le prix en fonction d'une variable, souvent le temps ou la quantité. 2. Supposon

1. **State the problem:** Liam's weekly train fare has increased by 8%, and he now pays an extra 2.20 every week. We need to find his new weekly train fare. 2. **Identify the varia

1. The problem is to solve the given equation or expression (not specified by the user). 2. Since no specific problem was provided, I will demonstrate a general solving approach: i

1. **Problem statement:** Given the piecewise function $$h(x) = \begin{cases} -2x + 4 & \text{if } x \leq 1 \\ x^2 + 1 & \text{if } x > 1 \end{cases}$$

1. Bài toán yêu cầu tìm giá bán $x$ để lợi nhuận $P(x) = (x-50)(120-x)$ đạt giá trị lớn nhất. 2. Hàm lợi nhuận là một hàm bậc hai dạng tích, ta có thể khai triển để dễ xử lý:

1. **Problem statement:** We have a piecewise function $$h(x) = \begin{cases} -2x + 4 & \text{if } x \leq 1 \\ x^2 + 1 & \text{if } x > 1 \end{cases}$$

1. Bài toán yêu cầu tìm giá bán $x$ để hàm lợi nhuận $P(x) = -x^2 + 50x - 20$ đạt giá trị lớn nhất. 2. Đây là hàm bậc hai có hệ số $a = -1 < 0$, nên đồ thị là parabol úp xuống và g

1. Bài toán yêu cầu lập hàm chi phí sản xuất theo thời gian $t$ dựa trên các hàm đã cho. 2. Ta có hàm tổng chi phí sản xuất theo số lượng sản phẩm $q$ là $$C(q) = q + 90$$

1. **State the problem:** Solve the equation $11^{-1}x = 10$ for $x$. 2. **Recall the rule:** $11^{-1}$ means the reciprocal of 11, so $11^{-1} = \frac{1}{11}$.

1. The problem is to verify if $$\sqrt[6]{\frac{9}{16}a^2b^4} = \sqrt[3]{\frac{3}{4}ab^2}$$ is correct. 2. Recall the rule for radicals: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$.

1. **State the problem:** Solve the equation $11^{-1}x = 10$ for $x$. 2. **Recall the rule:** $11^{-1}$ means the reciprocal of 11, so $11^{-1} = \frac{1}{11}$.