🧮 algebra

1. **Problem Statement:** Find the inverse of an exponential function, typically given as $y = a^x$ where $a > 0$ and $a \neq 1$. 2. **Formula and Important Rules:** The inverse of

1. Let's start by understanding what an exponential function is. An exponential function has the form $$y = a^x$$ where $a$ is a positive constant not equal to 1, and $x$ is the ex

1. **State the problem:** Expand and simplify the expression $$(c - 4)(c + 6) - (c + 2)(c - 3)$$. 2. **Recall the distributive property:** To expand products of binomials, use the

1. The problem is incomplete as only "A=[" is provided without further information. 2. To solve or analyze a matrix or set, we need the full elements or context.

1. **State the problem:** We have a binary operation defined as $a * b = a + b - 2$. We are asked to find the inverse of 5 under this operation, given that the identity element is

1. **State the problem:** We are given the quadratic function $f(x) = 3x^2 + 5x - 7$. 2. **Express $f(x)$ in the form $a(x+h)^2 + k$ (completing the square):**

1. **State the problem:** Orlando missed 5 out of 40 questions on the test. We need to find the percentage of questions he got right. 2. **Formula:** Percentage of correct answers

1. **Problem:** Factorize completely the expressions given. 2. **Part a:** Factorize $6y^2 - 18xy$.

1. Masala: Parabolaning uchi $(-1, -4)$ nuqtada joylashgan. $f(x) = ax^2 + bx + c$ tenglamada $a - b + c$ ni toping. Parabola uchi koordinatalari $x_0 = -\frac{b}{2a}$ va $y_0 = f(

1. Muammo: Parabolani $b(-3;4)$ vektor bo'yicha parallel ko'chirish natijasida hosil bo'lgan parabolaning tenglamasini toping. Formulasi: Agar $y=f(x)$ parabola $b(h;k)$ vektor bo'

1. The problem is to analyze and graph the piecewise function: $$Y = \begin{cases} -2x^2 + x - 5 & \text{if } x < -2 \\ |x - 4| & \text{if } x \geq 2 \end{cases}$$

1. مسئله را بیان می‌کنیم: الف) ضابطه جبری تابع داده شده را بیابید.

1. مسئله را بیان می‌کنیم: (ب) مقدار ماکزیمم یا مینیمم تابع $$f(x) = -(x + 1)^2 - 3$$ را پیدا کنید.

1. The problem is to find the value of $a - b + c$ given a parabola with vertex at $(-1, -4)$ and equation $f(x) = ax^2 + bx + c$. 2. The vertex formula for a parabola is $x_0 = -\

1. Muammo: Parabolaning uchi $(-1; -4)$ nuqtada joylashgan va $f(x) = ax^2 + bx + c$ tenglama berilgan. $a - b + c$ ni toping. 2. Parabolaning uchi koordinatalari $x_0 = -\frac{b}{

1. Muammo: Parabolani $y = 3x^2 + 2x - 7$ dan $b(-3;4)$ vektor bo'yicha parallel ko'chirish natijasida hosil bo'lgan parabolaning tenglamasini toping. 2. Parallel ko'chirishda yang

1. **State the problem:** Solve the exponential equation $$2^{2x+1} = 3^{x+1}$$. 2. **Recall the properties of exponents:** For any base $a > 0$, $a^{m+n} = a^m \cdot a^n$ and if $

1. **State the problem:** Solve the equation $$2x + \frac{1 - x}{4} = 3$$ and fully reduce the answer. 2. **Identify the formula and rules:** To solve linear equations with fractio

1. **State the problem:** Solve the equation $$5x - (12 + x) = -13$$ and simplify the answer fully. 2. **Apply the distributive property:** Remove the parentheses by distributing t

1. **Problem Statement:** We have a sequence of shapes with a number of tiles depending on $n$. The given values are $n=4, 8, 7, 5$ with corresponding tile counts $4, 8, 7, 5$. We

1. **State the problem:** We have a pattern of L-shaped figures made of tiles for values $n=1, 2, 3$. We want to find the number of tiles when $n=4$ and express the number of tiles