🧮 algebra

1. **State the problem:** We are given the system of equations:

1. **State the problem:** Find the value of $\theta$ that minimizes the function $$f(\theta) = (150\sin\theta + 183.384)(253.8 - 150\cos\theta).$$ 2. **Rewrite the function:**

1. **State the problem:** We need to find the factored form of $(-2ab)^3$. 2. **Recall the formula:** The cube of a product is the product of the cubes, i.e., $ (xyz)^3 = x^3 y^3 z

1. **State the problem:** Simplify the expression $(-2ab)^3$ with a positive exponent. 2. **Formula and rules:** When raising a product to a power, use the rule $$(xy)^n = x^n y^n$

1. **Stating the problem:** We want to understand the geometry of a straight line in algebra. 2. **Formula used:** The general equation of a straight line in the Cartesian plane is

1. The problem is to express a linear equation in the form $y = Mx + c$ where $M$ is the slope and $c$ is the y-intercept. 2. The general form of a linear equation is $y = Mx + c$.

1. The problem is to understand and analyze the quadratic function given by the formula $y = a(x+b)^2$. 2. This is a quadratic function in vertex form, where $a$ controls the width

1. **State the problem:** Solve the quadratic equation $a^2 = 9a + 14$ and then simplify the expression $2a^2 - 2a - 4$ using the solution. 2. **Rewrite the equation:** Move all te

1. The problem is to solve the expression $14c$ for a given value of $c$ or to simplify it if possible. 2. The expression $14c$ means $14$ multiplied by $c$. Without a specific val

1. **State the problem:** Solve the equation $12x = -11x - 20$ for $x$. 2. **Write down the equation:**

1. Таны өгсөн асуудал: Танкан дахь эхний усны хэмжээг $x$ литр гэж тэмдэглэе. 2. Асуудлын нөхцөлүүдийг томъёолох:

1. Асуудлыг тодорхойлж эхэлье: Өглөөний температур -5.4 градус Цельс байсан. 2. Өдрийн турш температур 12.8 градусаар нэмэгдсэн.

1. Асуудлыг тодорхойлж эхэлцгээе: Туршилтын төгсгөлд 0.85 литр химийн уусмал үлдсэн. 2. Туршилтын явцад 1.25 литр уусмалыг нэмж, 2.55 литрийг ашигласан гэж өгөгдсөн.

1. Problema ne oferă o lege de compoziție definită prin formula: $$x \bullet y = -\frac{1}{4}(x+1)(y+1) - 1$$

1. **State the problem:** We are given the function $h(x) = 2^x$ and the inverse function value $h^{-1}(x) = 5$. We need to find the value of $x$ such that $h^{-1}(x) = 5$. 2. **Re

1. **Stating the problem:** We want to transform the expression for $k_t^{E, \ell}$ given in the first formula (picture 1) into the simplified form shown in the third formula (pict

1. Diberikan fungsi kuadrat $f(x) = ax^2 - 4x + a - 2$ dengan nilai minimum 1. 2. Fungsi kuadrat $f(x) = ax^2 + bx + c$ memiliki nilai minimum jika $a > 0$ dan nilai minimumnya ada

1. **State the problem:** We need to simplify and understand the function $f(x) = 7\sqrt{(2x^2 + 6x)^3}$. 2. **Recall the formula and rules:** The square root of a power can be rew

1. **State the problem:** We are asked to sketch the graph of the function $$f(x) = (x - 1)^2 (x + 1)^3$$.

1. **State the problem:** Find the slope of the line passing through the points $(4,3)$ and $(5,8)$. 2. **Formula:** The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is

1. **State the problem:** Calculate the value of the expression $D=2-(-1)+2-(-2)$. 2. **Recall the rule:** Subtracting a negative number is equivalent to adding its positive counte