🧮 algebra

1. **Stating the problem:** The price of a holiday increases by 20%, which adds 240 to the original price. We need to find the original price before the increase. 2. **Formula and

1. **Stating the problem:** Solve the equation $x + 9 = 12$ for $x$. 2. **Formula and rules:** To isolate $x$, we use the rule that whatever we do to one side of the equation, we m

1. The problem involves an expression where the exponent is a square root in the denominator. 2. Let's consider a general expression of the form $$y = a^{\frac{1}{\sqrt{b}}}$$ wher

1. **State the problem:** Find the discriminant of the quadratic expression derived from $5^2 - 4(q)(-q + 3)$, which simplifies to $4q^2 - 12q + 25$. 2. **Recall the discriminant f

1. **State the problem:** Simplify the expression $5^2 - 4(q)(-q + 3)$. 2. **Recall the order of operations:** Calculate exponents first, then multiplication, and finally addition

1. **State the problem:** We want to prove that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}$$

1. Let's clarify the problem: We have an equation where powers of five are added to the denominators, not multiplied, with the numerators. 2. Suppose the equation is of the form $\

1. **State the problem:** Prove by induction that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}

1. Risolviamo l'equazione data. 2. Applichiamo le regole algebriche per isolare la variabile.

1. **State the problem:** Divide the polynomial $2a^3 - a - 1$ by the binomial $2a + 3$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We

1. **State the problem:** We want to prove by induction that the infinite nested radical expression $$x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$$ satisfies $$x < 2$$. 2. **Define

1. The problem is to simplify the expression $\frac{7n}{\epsilon} \div 5$. 2. Division by a number is the same as multiplication by its reciprocal, so we rewrite the expression as:

1. Das Problem lautet: Berechne den Ausdruck $\frac{7n}{\epsilon} + \frac{1}{5}$.\n\n2. Die Formel für die Addition von zwei Brüchen mit unterschiedlichen Nennern ist: $$\frac{a}{b

1. The problem is to simplify an expression and write it using only positive exponents. 2. The general rule is that any negative exponent $a^{-n}$ can be rewritten as $\frac{1}{a^n

1. **State the problem:** Simplify the algebraic expressions and write the answers with positive exponents only. 2. **Expression 1:** $4q^2 \times 4p^4 q^{-3}$

1. The problem involves analyzing the polynomial function $$f(x) = (x - 3)(x + \sqrt{5})(x - \sqrt{5})$$ and understanding its properties. 2. First, expand the function by multiply

1. **State the problem:** Simplify the expression $ (6b^2 - 4b^4) - (7b^2 - b^4) $. 2. **Rewrite the expression by distributing the minus sign:**

1. **State the problem:** Simplify the expression $$(1 - 4x^3) - (5 - 3x^3)$$. 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to each

1. **State the problem:** Find the points of intersection of the two parabolas given by the equations: $$y = x^2 - 2x + 2$$

1. **State the problem:** We are given that Jiahao and Raju have stamps in the ratio 3 : 1.

1. **Stating the problem:** Sam's savings and Ali's savings relate as follows: $\frac{3}{5}$ of Sam's savings equals $\frac{1}{4}$ of Ali's savings.