🧮 algebra

1. **State the problem:** We have a torpedo depth modeled by a quadratic pattern with depths at 1, 2, and 3 seconds given. We need to find the depth at 5 seconds, verify the formul

1. **State the problem:** We are given two functions $f(x) = 3x - 2$ and $g(x) = 2x + k$. We need to find the value of $k$ such that the composition $f \circ g = g \circ f$. 2. **W

1. Solve $(x+5)(x-2)=0$. Use the zero product property: $x+5=0$ or $x-2=0$.

1. The problem asks to find the compositions $f \circ g$ and $g \circ f$ where $f(x) = 2x + 1$ and $g(x) = x^2 - 2$. 2. First, find $f \circ g$, which means $f(g(x))$.

1. **State the problem:** We are given the function $$S(t) = \frac{1}{2}gt^2 + at + b$$ where $g$ is the acceleration due to gravity, and $a$, $b$ are constants. We need to check i

1. **State the problem:** We need to show that the function $f: \mathbb{R} \setminus \{0\} \to \mathbb{R} \setminus \{0\}$ defined by $f(x) = \frac{1}{x}$ is a bijection. 2. **Show

1. The problem asks to find the domain and range of the function $f(x) = x^2 + 1$. 2. The domain of a function is the set of all possible input values ($x$) for which the function

1. **Stating the problem:** Solve the equation $$\frac{1\frac{1}{2}p}{2} - 1 = \frac{3x}{3} + 6$$. 2. **Rewrite mixed number:** Convert $1\frac{1}{2}$ to an improper fraction: $$1\

1. **State the problem:** Find all unordered pairs of integers $(x,y)$ such that $$x^2 - y^2 = 75.$$ 2. **Rewrite using difference of squares:** We factor the left side as $$x^2 -

1. **State the problem:** Find the equation of the normal to the circle given by $$x^2 + y^2 - 2x - 4y - 12 = 0$$ at the point $$(-3, 5)$$. 2. **Rewrite the circle equation in stan

1. The problem is to find all unordered pairs of integers $(x,y)$ such that $$x^2 - y^2 = 75.$$\n\n2. Rewrite the equation using the difference of squares factorization: $$x^2 - y^

1. The problem is to find the number of unordered pairs $(x,y)$ where both $x$ and $y$ are positive and satisfy the equation $$x^2 - y^2 = 75.$$\n\n2. Rewrite the equation using th

1. The problem is to find the number of positive integer pairs $(x,y)$ such that $x^2 - y^2 = 49$. 2. Rewrite the equation using the difference of squares factorization:

1. The problem is to find the number of unordered pairs $(x,y)$ where $x$ and $y$ are integers satisfying the equation $$x^2 - y^2 = 49.$$ 2. Rewrite the equation using the differe

1. The problem is to analyze the equation of the form $$x^2 - y^2 = 49$$. 2. This equation represents a hyperbola because it is in the form $$x^2/a^2 - y^2/b^2 = 1$$ where $$a^2 =

1. **State the problem:** Five people went to a restaurant. Four of them each spent $12 less than the average amount spent by all five together, and the fifth person spent $18 less

1. The problem asks to analyze the conditions when $x,y$ are positive and when $x,y$ are real integers. 2. For the case when $x,y$ are positive, we consider $x>0$ and $y>0$.

1. The problem is to understand how the total number of pairs is calculated when given points (-7,0) and (7,0). 2. Each point represents a pair of coordinates in the form $(x,y)$.

1. The problem states: Find the number of unordered pairs $(x,y)$ such that $x^2 - y^2 = 49$. 2. Rewrite the equation using the difference of squares factorization:

1. The problem is to find the square root of 49. 2. Recall that the square root of a number $x$ is a value $y$ such that $y^2 = x$.

1. Vi starter med å skrive oppgaven: Løs likningen $$4x + (3x + 1) = 6 - 2(x + 1)$$. 2. Først fjerner vi parentesene på begge sider. På venstre side er det bare en parentes uten fo