🧮 algebra

1. **State the problem:** Prove that the quadratic equation $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a > 0$, has two real distinct solutions if and only if $b^2 >

1. **State the problem:** We are given several exponential functions and need to find for each:

1. **State the problem:** Determine which tables represent exponential functions and analyze given exponential functions for asymptotes, y-intercepts, domain, and range. 2. **Check

1. **State the problem:** Find the equation of the line passing through the point $(-5, 12)$ and perpendicular to the line given by $$y = \frac{2}{3}x + 14.$$ 2. **Recall the formu

1. **State the problem:** Find the center and radius of the circle given by the equation $$(x + 3)^2 + (y - 5)^2 = 16$$. 2. **Recall the standard form of a circle's equation:**

1. **State the problem:** Prove by contradiction that a polynomial equation with integer coefficients $$x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0$$

1. **State the problem:** Divide the polynomial $2x^3 - 9x^2 + x + 14$ by the binomial $x - 2$ using polynomial long division. 2. **Formula and rules:** Polynomial division is simi

1. **State the problem:** Divide the polynomial $2x^3 + 9x^2 + x + 14$ by the binomial $x - 2$ using polynomial long division. 2. **Formula and rules:** Polynomial division is simi

1. **State the problem:** Multiply the two matrices $$\begin{bmatrix}4 & 1 & 2 \\ \cdot & 3 & \end{bmatrix} \times \begin{bmatrix}2 & 3 & 8 & 4\end{bmatrix}$$

1. **State the problem:** Factor the quadratic expression $$x^2 - 3x - 4$$. 2. **Recall the factoring formula:** For a quadratic expression $$ax^2 + bx + c$$, we look for two numbe

1. **State the problem:** Factor the quadratic expression $$x^2 - 10x + 25$$. 2. **Recall the factoring formula:** A quadratic expression of the form $$x^2 + bx + c$$ can be factor

1. **State the problem:** Factor the quadratic expression $x^2 + 4x + 3$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** We are given two functions $f(m) = 8m + 7$ and $g(m) = 5m - 6$. We need to find the values of the compositions $(f \circ g)(-3)$ and $(g \circ f)(-3)$. 2.

1. The problem asks to find the values of the compositions $(f \circ g)(-6)$ and $(g \circ f)(-6)$ given the functions $f(y) = 3y - 8$ and $g(y) = 6y + 2$. 2. Recall that the compo

1. Problemet är att lösa ekvationen $$8\left(\frac{1}{x}\right) + 8\left(\frac{1}{x}\right) + 8\left(\frac{1}{x}\right) + 8\left(\frac{1}{x}\right) + 8\left(\frac{1}{x}\right) = 10

1. The problem is to understand the function $f(x) = \sqrt{}$, which appears incomplete as the square root has no expression inside. 2. The square root function is defined as $f(x)

1. The problem is to understand and evaluate the function given as $f(x) = \sqrt{}$ with no expression inside the square root. 2. The square root function $\sqrt{x}$ requires a num

1. Énoncé du problème : Résoudre le système d'équations linéaires donné. 2. Formules et règles importantes : Pour résoudre un système de deux équations linéaires à deux inconnues,

1. Énonçons le problème : Résoudre le système d'équations linéaires donné. 2. Le système est :

1. **Stating the problem:** Solve the inequality $$3x + 5 - 4x < 8x + 3 + 2x$$. 2. **Formula and rules:** To solve linear inequalities, we first simplify both sides, then isolate t

1. **State the problem:** We need to find the value of $y$ when $x=10.5$ for a piecewise function defined as: $$y=\begin{cases}-x + a, & 0 \leq x \leq 9 \\ b - x + c, & 9 \leq x \l