🧮 algebra

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 $11^{-1} = \frac{1}{11}$.

1. Stating the problem: Simplify the expression $$\sqrt[6]{\frac{9}{16}a^2b^4}$$. 2. Formula and rules: The sixth root of a product is the product of the sixth roots: $$\sqrt[6]{xy

1. The problem is to analyze the function $$f(x) = x^3 - 9$$. 2. This is a cubic function where the general form is $$f(x) = ax^3 + bx^2 + cx + d$$. Here, $$a=1$$, $$b=0$$, $$c=0$$

1. The problem is to simplify the expression $$\sqrt[8]{16a^8b^4}$$ and verify if it equals $$\sqrt{2a^2b}$$. 2. Recall the rule for radicals: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$.

1. **Problem Statement:** Revise and practice problems on Algebraic Expressions and Identities covering addition, subtraction, multiplication, and algebraic identities. 2. **Key Co

1. The problem is to evaluate $\frac{1}{0}$. 2. Division by zero is undefined in mathematics because there is no number that you can multiply by 0 to get 1.

1. The problem is to evaluate the expression $\frac{2}{0}$. 2. Division by zero is undefined in mathematics because dividing by zero does not produce a finite or meaningful result.

1. **State the problem:** Given the polynomial $$-x^2 + 8x + 9$$ with zeroes $$\alpha$$ and $$\beta$$ where $$\alpha > \beta$$, find the value of $$\alpha - \beta$$. 2. **Recall th