🧮 algebra

1. **State the problem:** Solve the equation $x^2 + 1 = 0$ for $x$. 2. **Recall the formula:** To solve quadratic equations of the form $ax^2 + bx + c = 0$, we use the quadratic fo

1. مسئله را بیان می‌کنیم: می‌خواهیم ترکیب توابع (gf) \circ (fg)، (gf) \circ (gf) و (fg) \circ (fg) را روی زوج مرتب (m,n) بررسی کنیم که شامل عبارات لگاریتمی و کسرهایی با مخرج‌هایی ب

1. **State the problem:** Solve the equation $x^2 + 1 = 5$ for $x$. 2. **Use the formula:** To solve for $x$, isolate $x^2$ by subtracting 1 from both sides:

1. **State the problem:** Solve the equation $$2x + 3 = 11$$ for $x$. 2. **Formula and rules:** To solve a linear equation, isolate the variable on one side by performing inverse o

1. Problem statement: Determine the correct expression among the options for $\log_a x$. 2. Formula: Use the change of base formula.

1. **State the problem:** We need to find the numbers to replace the boxes in the equation: $$\square x(x+3) + 9x(2x+1) = 5x(\square x + 6)$$

1. Problem Q10: You have black and white socks in the ratio 4:3. How many socks must you pull to be certain of having a single pair of the same color? 2. To solve this, use the Pig

1. The problem asks for the smallest possible product of two numbers that round to 20 and 300 respectively when rounded to 1 significant figure. 2. When rounding to 1 significant f

1. **Stating the problem:** We are given that the length of a string, $L$, is 30 cm when rounded to the nearest 10 cm.

1. **Problem Statement:** Prove by mathematical induction that $$3 + 3^2 + 3^3 + \cdots + 3^n = 0.5(3^{n+1} - 3)$$ for all positive integers $n$. 2. **Base Case:** Check for $n=1$.

1. **Stating the problem:** Solve the linear equation $x + 4y = 2$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation. Thi

1. **State the problem:** Calculate the total amounts of tomatoes, onions, and cooking oil needed for beef and beans for two occasions combined, then find the total cost for each,

1. **Énoncé du problème :** Déterminer l'ensemble de définition $D_f$ de la fonction $f$ dans chacun des cas suivants : a) $f(x) = \ln(x-1)$

1. **Stating the problem:** We want to disprove the statement: "If $a,b \notin \mathbb{Q}$ (irrational numbers), then $a^b \notin \mathbb{Q}$ (irrational)." 2. **Understanding the

1. The problem is to simplify the expression $$\frac{(-a)^{41}}{a^5}$$. 2. We use the laws of exponents: $$\frac{x^m}{x^n} = x^{m-n}$$ and $$(-a)^n = (-1)^n a^n$$.

1. The problem is to simplify the expression $$\frac{(40 - a)^5}{a^5}$$. 2. We use the property of exponents that states $$\frac{x^m}{y^m} = \left(\frac{x}{y}\right)^m$$.

1. The problem is to understand the expression $a$ to the power $b$, which is written as $a^b$. 2. The formula for exponentiation is $a^b = \underbrace{a \times a \times \cdots \ti

1. **Stating the problem:** We want to prove that if $a,b \notin \mathbb{Q}$ (i.e., $a$ and $b$ are irrational numbers), then the product $ab$ is also irrational, i.e., $ab \notin

1. **State the problem:** Find the zeros of the function $$f(x) = 2x^3 + 5x^2 - 11x + 4$$. 2. **Formula and rules:** To find zeros, solve $$f(x) = 0$$. For cubic polynomials, try r

1. The problem is to simplify the sum of three binary numbers: $1101_2 + 101_2 + 11_2$ and convert the result to decimal. 2. Recall that binary numbers are base 2, and to convert a

1. **State the problem:** Find the rational zeros of the polynomial $$f(x) = x^3 - 5x^2 + 2x + 1$$ using the Rational Zero Theorem. 2. **Recall the Rational Zero Theorem:** Possibl