🧮 algebra

1. **State the problem:** Solve the equation $\log_2(24 - 2^x) = x + 1$ for $x$. 2. **Rewrite the logarithmic equation in exponential form:**

1. **State the problem:** Solve the equation $\log_2(24 - 2^x) = x + 1$ for $x$.\n\n2. **Rewrite the logarithmic equation in exponential form:** Recall that $\log_b(a) = c$ means $

1. **State the problem:** We are given the quadratic equation $kx^2 + 5x = -7$ and need to find all values of $k$ such that the equation has two distinct real solutions. 2. **Rewri

1. **State the problem:** Simplify the expression $$2 - \frac{x + 2}{x - 3} - \frac{x - 6}{x + 3}$$ into a single fraction of the form $$\frac{ax + b}{x^2 - 9}$$ where $a$ and $b$

1. Énoncé : Trouver les coefficients $u$ et $v$ tels que $20u + 11v = 1$ en utilisant l'algorithme d'Euclide. 2. Appliquer l'algorithme d'Euclide pour calculer le PGCD de 20 et 11

1. Stating the problem: Simplify the expression $\frac{5}{6} - \left(\frac{1}{3} + \frac{1}{6}\right)$.\n\n2. First, simplify inside the parentheses: add $\frac{1}{3}$ and $\frac{1

1. Stating the problem: Simplify the expression $\frac{3}{4} + \left(\frac{1}{2} \times \frac{2}{3}\right)$.\n\n2. Multiply the fractions inside the parentheses: $$\frac{1}{2} \tim

1. Stating the problem: Simplify the expression $\frac{3}{4} + \left(\frac{1}{2} \times \frac{2}{3}\right)$.\n\n2. Calculate the multiplication inside the parentheses: $\frac{1}{2}

1. The problem is to evaluate the expression $\left(1 \frac{1}{2} + \frac{3}{4}\right) \div \frac{1}{4}$.\n\n2. Convert the mixed number $1 \frac{1}{2}$ to an improper fraction: $1

1. **State the problem:** We are given the quadratic equation $$kx^2 + 5x = -7$$ and need to find all values of the constant $$k$$ such that the equation has two distinct real solu

1. **State the problem:** Calculate $\left(1 \frac{1}{2} + \frac{3}{4}\right) \div \frac{1}{4}$. 2. **Convert mixed number to improper fraction:** $1 \frac{1}{2} = \frac{3}{2}$.

1. Problem statement: Analyze the function $f(x)=x^2$.\n2. Domain: The domain is all real numbers, written as $(-\infty,\infty)$, because squaring is defined for every real input.\

1. **State the problem:** Simplify the expression $$\frac{1}{6x^{2} + 7x - 5} \div \frac{1}{4x^{2} -1}$$ into the form $$\frac{ax+b}{cx+d}$$ where $a, b, c,$ and $d$ are integers.

1. **State the problem:** Solve the quadratic equation $$10x^2 + 3x = 0$$ for $x$. 2. **Factor the equation:** Factor out the common term $x$:

1. **State the problem:** Solve the equation $$\frac{1}{5-x} + \frac{2}{1+x} = 1$$ for $x$. 2. **Find a common denominator:** The denominators are $5-x$ and $1+x$. The common denom

1. **State the problem:** Solve the quadratic equation $$10x^2 - 4x = 0$$. 2. **Factor the equation:** Factor out the common term $2x$:

1. **State the problem:** Solve the quadratic equation $$4x^2 - 8x = 0$$ for $$x$$. 2. **Factor the equation:** Factor out the greatest common factor, which is $$4x$$:

1. **State the problem:** Solve the quadratic equation $$x^{2} - 5x = 0$$ for $x$. 2. **Factor the equation:** Factor out the common term $x$:

1. **State the problem:** Solve the quadratic equation $$x^2 - 5x = 0$$ for $$x$$. 2. **Factor the equation:** Factor out the common term $$x$$:

1. **Solving the equations:** a) Solve $5x - 3 = 7$:

1. The problem is to analyze the function $f(x) = x^2 + 2x + 3$. 2. First, identify the type of function: this is a quadratic function, which graphs as a parabola.