🧮 algebra

1. **State the problem:** Simplify the expression $\frac{65}{x}$ or understand its meaning. 2. **Formula and rules:** The expression $\frac{65}{x}$ represents division where 65 is

1. **State the problem:** Solve the two-step equation $$65x + 18 = 31$$. 2. **Understand the goal:** We want to isolate $x$ on one side of the equation to find its value.

1. সমস্যাটি হলো: 44% কর্মী কফি পছন্দ করে, 28% চা পছন্দ করে, এবং 40 জন কর্মী কেউ পছন্দ করে না। মোট কর্মী সংখ্যা কত? 2. আমরা জানি, মোট কর্মী সংখ্যা $N$ হলে, কফি পছন্দকারীর সংখ্যা $0.

1. **State the problem:** Solve the system of linear equations: $$4x + y = -114$$

1. **Problem 12:** Identify the linear system modeled by the balance scales. Given: Each small square represents 2 kg.

1. **State the problem:** Solve the quadratic equation $12x^2 = 25$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero.

1. **State the problem:** Determine if the given pairs of functions are inverses. 2. **Recall the rule:** Two functions $f$ and $g$ are inverses if and only if $f(g(x)) = x$ and $g

1. The problem asks for the sum of the items from 1 to 12. 2. We use the formula for the sum of an arithmetic series: $$S_n = \frac{n}{2}(a_1 + a_n)$$ where $n$ is the number of te

1. We are asked to find the inverse of each given function and then graph both the function and its inverse. 2. To find the inverse of a function $f(x)$, we swap $x$ and $y$ in the

1. The problem is to multiply the expressions $3s^2$ and $8s^5$. 2. The formula for multiplying terms with the same base is $a^m \times a^n = a^{m+n}$.

1. Stating the problem: We need to find the principal amount given the interest earned, rate, and time. 2. Formula for simple interest: $$I = P \times r \times t$$ where $I$ is int

1. Let's state the problem: We want to understand why in the quadratic formula, the product of the denominators of the roots equals $a$ when solving $ax^2 + bx + c = 0$. 2. The qua

1. **Problem:** Given the quadratic equation $ax^2 + \alpha x + c = 0$ with roots in the ratio $p:q$, prove that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{c}{a}} = 0.$$ 2. **Step 1: Expre

1. **State the problem:** Solve the quadratic equation $$x^2 - 11x + 28 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions

1. **Problem statement:** The roots of the equation $ax^2 + cx + c = 0$ are in the ratio $p : q$. Prove that $$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0.$$\n

1. **State the problem:** Solve the system of linear equations by graphing: $$\begin{cases} x + y = 6 \\ x - y = 2 \end{cases}$$

1. The problem is to understand and explain the concept of an asymptote in mathematics. 2. An asymptote is a line that a graph of a function approaches but never touches or crosses

1. **Énoncé du problème :** Simplifier l'expression $A = \ln(e^3) + 2 \ln(3e) + \ln\left(\frac{1}{9}\right)$. 2. **Formules et règles importantes :**

1. **State the problem:** Determine if the given pairs of functions are inverses of each other. 2. **Recall the definition:** Two functions $f$ and $g$ are inverses if and only if

1. **Problem statement:** Given that $a^p = b^q = c^r$ and $a, b, c$ are in geometric progression (G.P.), prove that $p, q, r$ are in harmonic progression (H.P.). 2. **Recall defin

1. **State the problem:** Simplify the expression $3 \cdot \sqrt{\frac{1}{9}} \cdot \sqrt{3}$.\n\n2. **Recall the properties of square roots:** For any positive numbers $a$ and $b$