🧮 algebra

1. **State the problem:** Find the turning point (vertex) of the curve given by the quadratic function $$y = x^2 - 6x + 8$$ by completing the square. 2. **Rewrite the quadratic exp

1. We are asked to evaluate the expression $2 + \sqrt{x+4}$ for values of $x$ between $-3$ and $12$ inclusive. 2. First, check the expression inside the square root: $$x+4$$ It mus

1. The problem is to find the greatest length of a wooden scale which can measure 540 cm and 360 cm exactly. 2. This means the scale length must be the greatest common divisor (GCD

1. Problem: Find the greatest length of a wooden scale which can be used to measure 540 cm and 360 cm exactly. Step 1: We need to find the Greatest Common Divisor (GCD) of 540 and

1. **Stating the problem:** We want to factorize the expression $$x^4 + 4y^4$$. 2. Notice that $$x^4 + 4y^4$$ is a sum of squares: $$x^4 + (2y^2)^2$$.

1. State the problem: We want to factorize the expression $14x^{2} - 49x^{2}$. 2. Combine like terms: Since both terms have $x^{2}$, subtract the coefficients:

1. The problem involves subtracting two scalar multiples of vectors: $$\frac{3}{7}(2, 8a - 2, 1) - \frac{1}{9}(2, 7a - 3, 6)$$

1. **State the problem:** We need to factorize the quadratic expression $x^2 + 14x + 49$. 2. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c$ where $a=1$, $b

1. Énoncer les définitions du cours : 1) Une valeur propre \(\lambda\) d'un endomorphisme \(f\) est un scalaire tel qu'il existe un vecteur non nul \(v\) vérifiant \(f(v) = \lambda

1. Muammoni tushuntirish: berilgan ifodalar ustida amallarni bajarish kerak: 28. $1 \tfrac{3}{7} (2.8a - 2.1) - 1 \tfrac{1}{9} (2.7a - 3.6)$ ni hisoblaymiz.

1. Let's start by writing the expression clearly: $$\sqrt{2 - \frac{\sqrt{3}}{2}} \times \sqrt{2 + \frac{\sqrt{3}}{2}}$$ 2. We can use the property of square roots that $$\sqrt{a}

1. **State the problem**: Simplify the expression $$\sqrt{2 - \frac{\sqrt{3}}{2}} \times \sqrt{2 + \frac{\sqrt{3}}{2}}.$$\n\n2. **Recall the property of radicals**: The product of

1. The problem involves calculating simple interest, which is given by the formula $$SI = \frac{P \times R \times T}{100}$$ where $P$ is the principal, $R$ is the rate per annum, a

1. Stating the problem: We have a geometric progression (GP) where the third term is $\frac{9}{2}$ and the fifth term is $\frac{81}{8}$. We need to find: A. The common ratio $r$

1. We are given a geometric progression with terms $2$, $x$, $y$, and $250$. 2. In a geometric progression, the ratio between consecutive terms is constant. Let this common ratio b

1. The problem asks to find the product of 1 and \(\pi\).\n2. Recall that multiplying any number by 1 results in the same number.\n3. Therefore, \(1 \times \pi = \pi\).\n\nFinal an

1. **Stating the problem:** Simplify and evaluate the expression $$\left(\frac{2\sqrt{2} - \sqrt{5}}{\sqrt{3}}\right)^{2023} \times \left(\frac{2\sqrt{2} + \sqrt{5}}{\sqrt{3}}\righ

1. Given the matrix equation: $$\begin{pmatrix} 3 & 2 \\ 7 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ 8 \end{pmatrix}$$

1. The problem is to divide one polynomial by another using long division. 2. Start by dividing the leading term of the dividend by the leading term of the divisor to find the firs

1. The problem states that the $n^{th}$ term of a sequence is given by $$a_n=(-2)^{\frac{n}{2}}.$$ 2. We are asked to find the $2n^{th}$ term of the sequence, denoted as $a_{2n}$.

1. **Problem (a):** A hire tool firm's net return decreases by 10% annually. The current net gain is 400. Find the total of all future profits assuming the tool lasts forever. 2. S