🧮 algebra

1. **State the problem:** Given the equation $$\frac{A}{x+2} + \frac{B}{x-3} = \frac{5x}{(x+2)(x-3)}$$ find the values of constants $A$ and $B$. 2. **Formula and approach:** To sol

1. **State the problem:** Write the expression $3 \log 2 - \frac{1}{2} \log 16$ as a single logarithm. 2. **Recall logarithm rules:**

1. The problem is to find the inverse function $f^{-1}(x)$ of the function $f(x) = 1 + e^{x}$. 2. To find the inverse, we start by setting $y = f(x)$, so:

1. **State the problem:** Simplify the expression $$e^{2x+3} \left(e^{3x-5}\right)^2$$. 2. **Recall the exponent rules:** When multiplying expressions with the same base, add the e

1. **State the problem:** We need to find the value of $k$ such that $(x + 2)$ is a factor of the polynomial $x^3 + 2x^2 + kx + 6$. 2. **Recall the Factor Theorem:** If $(x + 2)$ i

1. **State the problem:** We are given a cubic polynomial $$x^3 + 2x^2 + kx + 6$$ and told that $$(x + 2)$$ is a factor. We need to find the value of $$k$$. 2. **Recall the Factor

1. **State the problem:** We need to find the value of $k$ such that $(x + 2)$ is a factor of the polynomial $$x^3 + 2x^2 + kx + 6.$$ 2. **Recall the Factor Theorem:** If $(x + 2)$

1. **State the problem:** Solve for $y$ in the equation $$4y^2 - 2 = 9y - 4$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** Solve for $y$ in the equation $$4y^2 - 2 = 9y - 4$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** Solve for $y$ in the equation $$4y^2 - 2 = 9y - 4$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** Solve for $y$ in the equation $$4y^2 - 2 = 9y - 4$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** We are given that $x = 8$ and some conditions involving $x$ and $y$. 2. **Translate the problem into equations:**

1. **Énoncé du problème :** Calculer la dérivée de la fonction $g(x) = x^3 + 3x - 4$ pour tout $x \in \mathbb{R}$.

1. **State the problem:** Simplify the expression $$\frac{(3x-4)(7x-3)}{(7x-3)(x+6)} \cdot \frac{(x+6)}{(3x-4)}$$. 2. **Recall the rule:** When multiplying fractions, multiply nume

1. **State the problem:** Simplify or factor the expression $$y^4 + y^2 - 2ay + 1 - a^2$$. 2. **Recall the formula:** Recognize that $$1 - a^2$$ can be written as $$(1 - a)(1 + a)$

1. **State the problem:** Factorise the expression $$(y+z-x)^3 + (z+x-y)^3 + (x+y-z)^3 = -24xyz$$ given that $$x + y + z = 0$$. 2. **Recall the identity:** There is a well-known fa

1. **State the problem:** Simplify or analyze the expression $x^2 + 3y^2 - z^2 + 2yz - 4xy$. 2. **Identify the terms:** The expression contains quadratic terms in $x$, $y$, and $z$

1. **State the problem:** We are given the expression $6x + 12$ and asked to analyze it. 2. **Identify the type of function:** The expression $6x + 12$ is a linear function, not a

1. **State the problem:** Simplify the expression $$(x+1)(x+3)(x+5)(x+7) + 15$$. 2. **Use the formula and rules:** We will first multiply the factors in pairs to simplify the expre

1. **State the problem:** Factorise the expression $$a^2 + 2ab - ac - 3b^2 + 5bc - 2c^2$$. 2. **Group terms:** Group the terms to make factorisation easier:

1. **Problem Statement:** Solve the equation $$x^2 - 5x + 6 = 0$$. 2. **Formula Used:** For quadratic equations of the form $$ax^2 + bx + c = 0$$, the solutions are given by the qu