🧮 algebra

1. **Problem statement:** We are given a rectangle whose length is twice its width and its perimeter is 54 cm.

1. The nature of roots of a quadratic equation $ax^2+bx+c=0$ depends on the discriminant $\Delta=b^2-4ac$. 2. If $\Delta>0$, there are two distinct real roots.

1. **State the problem:** Differentiate between rational and irrational numbers. 2. **Explanation:**

1. **Énoncé du problème**: Montrer que l'ensemble $$F = \{(x,y,z) \in \mathbb{R}^3 \mid x + 2y = 0\}$$ est un sous-espace vectoriel de $$\mathbb{R}^3$$ et donner une base de $$F$$.

1. Let's start by stating the general form of a quadratic equation: $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. To solve this quadratic equation,

1. Problem: Simplify $\frac{\sqrt{2}}{\sqrt{3}}$. Work: Multiply numerator and denominator by $\sqrt{3}$ to rationalize the denominator.

1. Let's start by clearly stating the problem: Solve the equation like a student. 2. Since the user did not specify an equation, assume a simple example: Solve for $x$ in the equat

1. সমস্যা বর্ণনা: গ) y^4 - 1 \over y^3 + y কে সবচেয়ে সরল রূপে প্রকাশ করতে হবে।

1. Problemi: Sa është 4% e 5600? Zgjidhje:

1. Find 4% of 5600. Step: $4\% = \frac{4}{100} = 0.04$,

1. Stating the problem: We need to verify if $$((x+y)(z^2+w^2))^2 = (x+y)^2 + 2(x+y)(z^2+w^2) + (z^2+w^2)^2$$ is true. 2. Expanding the left side: $$((x+y)(z^2+w^2))^2 = ((x+y)(z^2

1. The problem is to calculate $\frac{12}{5} \times 1.5$. 2. First, rewrite $1.5$ as a fraction: $1.5 = \frac{3}{2}$.

1. We are asked to compute the expression $-\frac{4}{3} : \frac{13}{9}$. This means we need to divide $-\frac{4}{3}$ by $\frac{13}{9}$. 2. Recall that dividing by a fraction is the

1. The problem is to solve the equation for $x$: $$\frac{x}{\frac{5}{7}} = -\frac{21}{20}.$$\n\n2. Rewrite the equation to isolate $x$: $$x \times \frac{7}{5} = -\frac{21}{20}.$$\n

1. Let's state the problem: Verify the given algebraic identity: $$((sx+y)(z^2+w^2))^2 = (s+y)^2 + 2(s+y)(z^2+w^2) + (z^2+w^2)^2$$

1. Stating the problem: Solve the equation $x + \frac{5}{4} = -\frac{1}{2}$ for $x$. 2. To isolate $x$, subtract $\frac{5}{4}$ from both sides:

1. **State the problem:** Calculate $-\frac{8}{15} + 0.6$. 2. **Convert 0.6 to a fraction:**

1. Stating the problem: Simplify the expression $$\frac{-8}{15} + 0$$. 2. Since adding zero to any number does not change its value, the expression remains $$\frac{-8}{15}$$.

1. Stating the problem: Solve the equation $x - \frac{3}{4} = \frac{2}{7}$ for $x$. 2. To isolate $x$, add $\frac{3}{4}$ to both sides of the equation:

1. Stating the problem: Calculate the product of $2.8$ and $-\frac{20}{7}$. 2. Express the multiplication explicitly:

1. The problem involves a function machine where the input is first transformed by multiplication by 5, then the output of this step is processed further. 2. Input to function mach