🧮 algebra

1. Statement of the problem: Factor the polynomial $x^2-1$ and find its roots. 2. Recognize that this is a difference of squares because $x^2-1^2=x^2-1$.

1. The problem states a parabola given by the equation $$Y = a(x-u)x^2 + 5$$ passing through points A(2,7) and B(ç,5). We need to find the constants $a$ and $u$ and verify if the p

1. **Problem Statement:** Given the Venn diagram with three sets "Music," "Art," and "Dance," and their overlapping student counts expressed algebraically, express the total number

1. **Stating the problem:** Solve and analyze the equation $$x^2 - 4x + 2y + y^2 - 11 = 0$$ for its geometric representation. 2. **Rewrite the equation:** Group $x$ and $y$ terms:

1. Stoiat zadachata: Da se nameri kakva stoinost treba da ima $k$, za da korenite na uravnenieto $$x^3 - kx^2 - x = 0$$ obrazuvat aritmetichna progresiq. 2. Razglejdame uravnenieto

1. Задачата ни е да намерим стойността на $k$, при която корените на уравнението $$3x^2 - kx - x = 0$$ образуват аритметична прогресия. 2. Първо опростяваме уравнението: $$3x^2 - k

1. The problem is to sketch the graph of the function $f(t)=5u(t-2)$, where $u(t)$ is the Heaviside step function. 2. Recall the Heaviside step function $u(t-a)$ is defined as 0 fo

1. **Stating the problem:** Three girls paid $100 each, totaling $300, for a motel room. The clerk realized the correct charge was $250, so $50 was returned via the attendant. The

1. **Stating the problem:** Three girls initially paid a total of $300 for a motel room ($100 each). Later, they were supposed to be charged only $250, so the clerk gives $50 to th

1. The problem is to find the x- and y-intercepts of a function and identify extrema (maxima or minima). 2. The function described is not specified, so let's consider a generic exa

1. The problem is to solve the equation $$3x^2 - 12x + 9 = 0$$. 2. First, we can simplify the equation by dividing all terms by 3:

1. The problem involves a number machine with two steps: first an operation \(An!\), then multiplication by 5, giving the output. 2. Given the input is 15 and the output is 30, we

1. The problem gives the number of dots for the first three terms: term 1 has 5 dots, term 2 has 8 dots, and term 3 has 11 dots. 2. We observe that the number of dots increases by

1. We are given a sequence where the 1st pattern has 5 dots, the 2nd has 8 dots, and the 3rd has 11 dots. We want to find an expression for the number of dots in the $n$th pattern.

1. The problem asks for a rule for the number of sticks in the nth pattern, given: 1st pattern has 5 sticks, 2nd pattern has 9 sticks, 3rd pattern has 13 sticks. 2. Observe the pat

1. **State the problem:** Find the geometric mean between the pairs of numbers given: I. 9 and 16

1. The problem presents a pattern of dots where the 1st pattern has 7 dots, the 2nd has 10 dots, and the 3rd has 13 dots. 2. Observe the differences between consecutive terms: $10

1. **Problem (a): Solve the indicial equation** $2^x + 2^{x-1} = 48$ 2. Rewrite $2^{x-1}$ as $2^x \cdot \frac{1}{2}$ since subtracting 1 from the exponent divides by 2.

1. **Simplify each algebraic expression:** i) Simplify $7a + 3b - 5a + b - 6$

1. **State the problem:** We need to find the first five terms and the tenth term of a geometric sequence where the first term $a=3$ and the common ratio $r=\frac{-1}{2}$. 2. **Rec

1. Solve for $x$ in each indicial equation. **a.** $2^x + 2^{x-1} = 48$