🧮 algebra

1. Problem: Mamy funkcję $f(x)$ i chcemy zrozumieć, co się dzieje z wykresem funkcji, gdy mamy wyrażenie $f(x+1) - 2$. 2. Reguły przesunięć wykresu funkcji:

1. **Problem:** The length of a rectangle is twice the width. The perimeter is 20 cm. Find the width. 2. **Formula:** The perimeter $P$ of a rectangle is given by $$P = 2(\text{len

1. **Problem Statement:** Evaluate the following matrix expressions and find the values of variables $x$, $y$, and $z$ given matrices $M$, $A$, and $B$.

1. **Problem:** Given the point (1, 2) lies on the graph of $y=2^x$, find the corresponding point on the graph of $y=3x^2 - 7x + 18$. 2. **Formula:** To find the corresponding poin

1. **State the problem:** We have revenue function $R(x) = 110x$ and cost function $C(x) = 60x + 7500$, where $x$ is the quantity sold.

1. **State the problem:** Joshua covers a floor of area 45 m² with tiles. He buys one box of tiles for every 1.5 m², each box costs 64, and he buys 5 bags of adhesive at 12 each. H

1. **State the problem:** We want to find an approximate root of the equation $$\sin(x) = x^2 - 1$$ using Newton's method. 2. **Rewrite the equation:** Define a function $$f(x) = \

1. Problem: If the point (3,4) lies on the graph of the equation $3y = ax - 7$, find the value of $a$. 2. Substitute $x=3$ and $y=4$ into the equation:

1. **State the problem:** Solve the equation $$\frac{5 - 2x}{6} = 3x - 4$$ for $x$. 2. **Multiply both sides by 6** to eliminate the denominator:

1. **Problem statement:** We are asked to determine the truth value of several statements about a function $f$ given only its variation table: $$\begin{array}{c|ccccc}

1. **State the problem:** Find the coefficient of $x$ in the expansion of $\left(5x - 3\right)\left(2x^2 - 3x + 1\right)$.\n\n2. **Write the expression to expand:** \n$$\left(5x -

1. **State the problem:** Evaluate the expression $$-5^2 - \left[1 - \frac{12}{(-2)^2}\right] + (-1)^3$$. 2. **Recall order of operations:** Exponents first, then multiplication/di

1. **State the problem:** Solve the inequality $\frac{2}{3x+6} < 0$. 2. **Rewrite the inequality:** The denominator is $3x+6$. The fraction is negative when numerator and denominat

1. **Problem statement:** Given the functions $f(x) = 5x + 1$ and $g(x) = \frac{x - 4}{2}$, find the value of $x$ if $fg(x) = 11$. 2. **Recall the composition of functions:**

1. Problem: Calculate $\frac{4}{5} + \frac{5}{6}$ and write the answer as a mixed number if appropriate. 2. Formula: To add fractions, find a common denominator, then add the numer

1. **Stating the problem:** We are given the quadratic equation $$16x^2 - 8x - 47 = 0$$ and told its roots are $$\frac{1}{4} + \sqrt{5}$$ and $$\frac{1}{4} - \sqrt{5}$$. We need to

1. **Илэрхийллийн утгыг олоорой:** $$9105 - 7865 + (2456 + 4895 - 3067) - 398$$ 2. **Тоо бодох үйлдлийг дараалалтай гүйцэтгэнэ:**

1. **Stating the problem:** Evaluate the summation $$\sum_{k=1}^5 (1 + k^3)$$. 2. **Recall the summation properties:** The sum of a sum is the sum of the sums, so

1. **State the problem:** Solve the equation $x + 2xy - y^2 = 2$ for $x$ in terms of $y$. 2. **Rewrite the equation:**

1. **State the problem:** Solve the equation $x + 2xy - y^2 = 2$ for $x$ in terms of $y$. 2. **Rewrite the equation:** The equation is $x + 2xy - y^2 = 2$.

1. **State the problem:** Simplify the function $f(x) = \frac{x^5 - 2x^3 - 8x^2}{x}$. 2. **Recall the rule:** When dividing a polynomial by $x$, divide each term in the numerator b