🧮 algebra

1. **State the problem:** We need to solve the inequality $$a (x^2 - x - a) \cdot (\sqrt{x^2 - 4x + 4} - a) > 0$$ for each value of the parameter $a$. 2. **Simplify the expression

1. **State the problem:** Solve the equation $$\frac{1}{2} \left(x + \frac{2}{3}\right) = \frac{13}{6}$$ for $x$. 2. **Use the distributive property:** Multiply both sides by 2 to

1. The problem is to solve for $x$ in the equation $2x + 3 = 7$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **State the problem:** We are given the second term of a geometric progression (GP) as 5, and we need to find the product of the first three terms. 2. **Recall the formula for t

1. Gegeben ist die Gleichung, die gelöst werden soll. 2. Wir lösen die Gleichung Schritt für Schritt.

1. The problem is to solve the equation shown in the screenshot (assuming it is a quadratic equation or similar algebraic expression). 2. Identify the type of equation and write do

1. The problem is to understand the function notation $d(t)$, which typically represents a function named $d$ with the variable $t$. 2. In mathematics, $d(t)$ means the value of th

1. El problema es encontrar el límite de la función $$d(t) = \frac{t^2 - 25}{t - 5}$$ cuando $$t$$ se acerca a 5. 2. La fórmula para límites que involucran una indeterminación $$\f

1. **State the problem:** We need to show that the line $y = 3x - 2k$ and the curve $y = x^2 - kx + 2$ intersect for all values of $k$. 2. **Set the equations equal to find interse

1. Planteamos el problema: calcular el límite $$\lim_{t \to 5} \frac{t^2 - 25}{t - 5}$$. 2. Observamos que el numerador es una diferencia de cuadrados: $$t^2 - 25 = (t - 5)(t + 5)$

1. **Problem statement:** Given the parabola $y = x^2 - 4$ and the line $y = mx + c$ with $m = -4$, find the values of $c$ for which the line intersects the parabola at two distinc

1. **State the problem:** We have a line with equation $y = mx + c$ and a curve with equation $xy = 16$. The line is tangent to the curve, and we need to express $m$ in terms of $c

1. **State the problem:** Simplify the expression $$\left(6x y^{\frac{1}{2}}\right)^{\frac{2}{4}} \left(8x y^{\frac{3}{2}}\right)^{-4}$$. 2. **Rewrite the exponents:** Note that $$

1. **State the problem:** We need to find an expression for the perimeter of the given cross-shaped figure. 2. **Identify the sides:** The figure has edges labeled with variables:

1. **State the problem:** Find how much longer the length of the rectangular garden is than the width.

1. The problem asks to find $(g - h)(x)$ where $g(x) = 5x^3 + 16x + 7$ and $h(x) = 2x^3 - 5x + 8$. 2. The formula for subtraction of functions is $(g - h)(x) = g(x) - h(x)$.

1. **State the problem:** We need to find the values of $m$ such that the line $y = mx - 6$ is tangent to the curve $y = x^2 - 4x + 3$. Also, find the coordinates of the tangent po

1. **State the problem:** We are given the circle equation $$x^2 + y^2 + 6x - 2y - 26 = 0$$ and the line equation $$y = kx - 5$$. We need to find the values of $$k$$ for which the

1. The problem is to determine if the equation $2x^4 + 6x^2 = 8x^8$ is true for all values of $x$. 2. To check this, we can try to simplify or rearrange the equation and analyze it

1. **State the problem:** Solve the equation $$2x^4 + 6x^2 = 8x^8$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem asks: 48 is what percent of 94? 2. To find what percent one number is of another, use the formula: