🧮 algebra

1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} 4x - 5, & x \leq 2 \\ 3x - 2x^2, & x > 2 \end{cases}$$

1. Задача: разложить выражение $x^4 - 27x$ на множители. 2. Формула и правила: для разложения на множители сначала вынесем общий множитель за скобки, затем применим формулы разложе

1. **State the problem:** We need to find the value of $f$ for the point $(5, f)$ on a line passing through $(-1, 9)$ with a gradient (slope) of $-2$. 2. **Formula for gradient:**

1. Задача: упростить выражение $49x^6 - 64$. 2. Формула: это разность квадратов, которая имеет вид $$a^2 - b^2 = (a - b)(a + b)$$.

1. The problem states that the quadratic expression $a x^{2} + b x + c$ can be rewritten as $$10 (x - 3)^{2} - 23$$ by completing the square. 2. Recall the formula for expanding a

1. Задача: упростить выражение $49x^2 - 64$. 2. Формула: это разность квадратов, которая упрощается по формуле $$a^2 - b^2 = (a - b)(a + b)$$.

1. **State the problem:** We have a multiplication grid with unknowns $a$, $b$, $c$, and $d$. The grid shows products of numbers in the top row and left column. 2. **Understand the

1. **State the problem:** Given the equation $\log x + \log 3 = y \log 6 + \log 6$, show that $x = y + 1$. 2. **Recall logarithm properties:**

1. **State the problem:** We need to calculate the percentage error when the predicted temperature is 25 and the actual temperature is 22. 2. **Formula for percentage error:**

1. **State the problem:** Find the gradient (slope) of the straight line passing through the points $(14, -6)$ and $(16, 10a)$. 2. **Formula for gradient:** The gradient $m$ of a l

1. **State the problem:** We want to rewrite the quadratic expression $2x^2 - 6x + 5$ in the form $a(x + b)^2 + c$, where $a$, $b$, and $c$ are numbers in simplest form. 2. **Formu

1. **State the problem:** Given the equation $\log x + x \log 3 = y \log 6 + \log 6$, show that $x = y + 1$. 2. **Recall logarithm properties:**

1. **Problem a:** Solve $\left(\frac{3}{2}\right)^{2 - x^2} = \frac{44}{9}$. 2. Take logarithm on both sides: $\log \left(\left(\frac{3}{2}\right)^{2 - x^2}\right) = \log \left(\fr

1. **Problem a:** Solve $\left(\frac{3}{2}\right)^{2 - x^2} = \frac{41}{9}$. - We recognize $\frac{41}{9}$ is not a simple power of $\frac{3}{2}$, so we take logarithms on both sid

1. The problem is to express the polynomial $9x^{10} - 15x^4 + 9$ in quadratic form if possible. 2. A quadratic form is generally an expression of the form $ax^2 + bx + c$ where th

1. **State the problem:** We need to rewrite the expression $9x^{10} - 15x^4 + 9$ in quadratic form. 2. **Understand quadratic form:** A quadratic form is an expression of the form

1. The problem is unclear as "take l x" does not specify a mathematical operation or expression. 2. Please provide a complete mathematical expression or clarify the operation you w

1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} x^2 + 1, & x < 1 \\ 1, & x = 1 \\ \sqrt{x + 3}, & x > 1 \end{cases}$$

1. **State the problem:** We are given two integers $x$ and $y$ such that their difference is $x - y = -18$. We want to find: a) The minimum product of the two numbers.

1. **State the problem:** We need to find a number $x$ such that $x > 344$ and $x$ rounds to 340 when rounded to 2 significant figures. 2. **Understand rounding to 2 significant fi

1. The problem is to solve the quadratic equation $$2x^2 - 3x - 2 = 0$$ for $x$. 2. We use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-3$, an