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  • Algebra Fundamentals
    Quadratic equations, functions, and graphing techniques.
    Updated 2 hours ago
  • Chemistry Basics
    Atomic structure, periodic table, and chemical bonding.
    Updated yesterday
  • Physics Formulas
    Key physics equations for motion, energy, and electricity.
    Updated 3 days ago
  • Advanced Mathematics
    Calculus, trigonometry, and complex numbers.
    Updated 1 week ago

Algebra Fundamentals

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x, with a ≠ 0:

\[ ax^2 + bx + c = 0 \]

The solutions to the quadratic equation are given by the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Example Problem

Solve the equation: \( 2x^2 - 4x - 6 = 0 \)

Solution:

Using the quadratic formula where a=2, b=-4, c=-6:

\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)} \] \[ x = \frac{4 \pm \sqrt{16 + 48}}{4} \] \[ x = \frac{4 \pm \sqrt{64}}{4} \] \[ x = \frac{4 \pm 8}{4} \]

So the solutions are:

\[ x = \frac{4 + 8}{4} = 3 \quad \text{and} \quad x = \frac{4 - 8}{4} = -1 \]

Chemical Reactions

Here's an example of a chemical reaction with a diagram:

\[ 2H_2 + O_2 \rightarrow 2H_2O \]

This shows the formation of water from hydrogen and oxygen gas.

Chemistry lab

Image: Laboratory equipment for chemical experiments

Physics Equations

Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass:

\[ F = ma \]

The kinetic energy of an object is given by:

\[ E_k = \frac{1}{2}mv^2 \]

Key Points to Remember

  • Quadratic equations always have the form \( ax^2 + bx + c = 0 \)
  • The discriminant \( b^2 - 4ac \) determines the nature of the roots
  • Chemical equations must be balanced to satisfy the law of conservation of mass
  • Force equals mass times acceleration according to Newton's second law