🧮 algebra

1. **State the problem:** We want to find the values of $a$ such that both roots of the quadratic equation $$(a-1)x^2 - 2x + 5 = 0$$ are greater than 2. 2. **Recall the quadratic f

1. The problem asks to solve for $x_0$ and find $f(2)$ given a function $f(x)$. 2. To solve for $x_0$, we need the equation or condition involving $x_0$. Since it is not provided,

1. **State the problem:** We want to find the values of $a$ such that both roots of the quadratic equation $$(a-1)x^2 - 2x + 5 = 0$$ are greater than 2. 2. **Recall the quadratic f

1. **State the problem:** Simplify the expression $$\frac{x^3 - 27}{x - 3}$$. 2. **Recall the formula:** The numerator is a difference of cubes, which can be factored using the for

1. समस्या: हमें \(\sqrt{x}\) का सही प्रतिनिधित्व करना है। 2. सूत्र: \(\sqrt{x}\) का अर्थ है \(x\) का वर्गमूल, यानी वह संख्या जो खुद से गुणा करने पर \(x\) दे।

1. **State the problem:** Simplify the expression $$\sqrt{13} + \frac{3}{\sqrt{13} - 3}$$. 2. **Recall the formula:** To simplify expressions with radicals in the denominator, mult

1. **State the problem:** Given $x=3+\sqrt{8}$, find the value of $\left(x-\frac{1}{x}\right)^2$. 2. **Recall the formula:** We want to evaluate $\left(x-\frac{1}{x}\right)^2 = x^2

1. **State the problem:** We have the function $$f(x) = x^4 - ax^3 + ax^2 - 2x + 1$$ with $$a = 2$$ and the factorization $$f(x) = (x - 1) \cdot g(x)$$. We want to find the values

1. The problem is to understand the expression $x = 3 + \sqrt{8}$ and simplify it if possible. 2. Recall that $\sqrt{8}$ means the square root of 8, which can be simplified by fact

1. **State the problem:** Simplify the expression $$\left(x-\frac{1}{x}\right)^2$$. 2. **Formula used:** The square of a binomial $$ (a-b)^2 = a^2 - 2ab + b^2 $$.

1. **State the problem:** We need to find the rate of change of the function $q = m + 2m - 1$ as $m$ varies. 2. **Simplify the function:** Combine like terms:

1. সমস্যাটি হলো: একটি আয়তাকার বাগানের ক্ষেত্রফল ৩০০ বর্গমিটার। এর প্রস্থ ১৫ মিটার হলে এর দৈর্ঘ্য কত হবে? 2. ক্ষেত্রফল (Area) সূত্র: $$\text{ক্ষেত্রফল} = \text{দৈর্ঘ্য} \times \tex

1. **Problem Statement:** Find the number that must be added to the polynomial $$2x^3 - 7x^2 + 2x$$ so that when the resulting polynomial is divided by $$2x - 3$$, the remainder is

1. **State the problem:** We want to find the values of $a$ such that both roots of the quadratic equation $$x^2-(2a+1)x+a^2-4a+3=0$$ are greater than 0. 2. **Recall the quadratic

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