🧮 algebra

1. **Stating the problem:** Given the equation $$x^4 + 1 = \frac{5}{8} x^2$$, find the values of: - $$x + \frac{1}{x}$$

1. **State the problem:** We need to find 65% of 15. 2. **Convert percentage to decimal:** 65% means 65 out of 100, which is written as $0.65$.

1. **State the problem:** Given functions $f(x) = x^2 - 2x$ and $g(x) = 2x + 3$, and the equation $f \circ g^{-1}(x) = 3$, find the value of $x$. 2. **Find the inverse of $g$: **

1. Let's start by defining a piecewise function. A piecewise function is defined by different expressions over different intervals of the domain. 2. Consider this example of a piec

1. The problem gives two functions: $$g(x) = 2x+2$$ and $$f \circ g(x) = 8x+13$$. We need to find $$x$$ such that $$g \circ f(x) = 20$$. 2. First, understand the notation: $$f \cir

1. **State the problem:** Mr. Sanchez buys $4\frac{3}{4}$ kg of fish to make Spanish sardines. Each can contains $\frac{1}{12}$ kg of fish. Each can is sold for 25 pesos. We need t

1. The problem is to find how much money Mr. Sanchez will earn after selling all the cans of Spanish sardines he can make from 43/4 kg of fish. Each can contains 1/12 kg of fish an

1. State the problem: We are given the set $S = \{(x,y) \mid x,y \in \mathbb{R} \text{ and } y = 2x - 2\}$, which represents points $(x,y)$ in the real plane satisfying the linear

1. **Problem restatement:** We want to find the value of $a$ such that the line $2x - 3y = 6$ passes through points $(9,-4)$ and $(6,a)$. 2. **Substitute first point into the line

1. First, let's clarify what the "simple method" means in general terms: it's a straightforward approach to solving a problem without complicated steps. 2. For example, if you're s

1. Stating the problem: We have functions $f(x) = ax + 5$ and functions $f \circ g(x) = 8x + 13$, and are given that $g \circ f(5) = 8a$. We need to find the value of $a$. 2. Expre

1. The problem states that the cube root of $a$ equals the square root of 5, i.e., $\sqrt[3]{a} = \sqrt{5}$. 2. We want to find the value of $a$. To do this, we can cube both sides

1. The problem states: Given the function $y = 4x + 2$, find its characteristics and interpret its meaning. 2. This is a linear function where $y$ depends on $x$ with a slope of 4

1. Diberikan persamaan: $$2(2^x - 1)x^2 + (2x^2 - 2)x = 2^{x+1} - 2$$

1. **Aljafna (Equations)** - **Title:** Algebraic Equations

1. **State the problem:** We need to find two costs related to the garden: the cost of repairing the garden floor and the cost of constructing a wall around it. 2. **Given data:**

1. **State the problem:** Simplify the expression $9 - 2\sqrt{14}$. 2. **Identify the terms:** Here, $9$ is a constant and $2\sqrt{14}$ is an irrational number. Since they are unli

1. Given polynomial $p(x) = mx^3 - 29x^2 + 39x + n$ with factor $3x - 1$ and remainder 6 when divided by $x - 1$. Step 1: Since $3x - 1$ is a factor, $p(\frac{1}{3}) = 0$. Substitu

1. Solve the equation $n - (3 + 2n) = -6$. Start by distributing and combining like terms:

1. **State the problem:** Solve the equation $$\sqrt[3]{x+1}+\sqrt[3]{-x+1}=2$$. 2. **Set variables:** Let $$a=\sqrt[3]{x+1}$$ and $$b=\sqrt[3]{-x+1}$$.

1. **Stating the problem:** The retailer buys a radio for rs. 225 and has overhead expenses of rs. 15. He sells the radio for rs. 300. We need to determine his profit.