Quadratic Equations and Their Solutions

1. Quadratic Equation
   A quadratic equation in is an equation of the form
   quadratic expression in , where all terms are moved to the left-hand side. A general quadratic equation is:
   where are constants, and
   
   
2. Solution (Root) of a Quadratic Equation
   The values of that satisfy the equation are called the solutions or roots of the quadratic equation.
   
   
3. Solving a Quadratic Equation
   Finding all the solutions of a quadratic equation is referred to as solving the quadratic equation.

If no restrictions are given on the values of , assume the solutions are real numbers.

Solving Quadratic Equations Using Factoring

1. Property of
   For two numbers or expressions and :
1. If , then or
2. If or , then


2. or implies one of three possibilities:
1. and
2. and
3. and


3. Steps to Solve a Quadratic Equation by Factoring
1. Arrange the equation as:
   
2. Factor the left-hand side:
   
3. Use the property of :
   or
4. Find the solutions:
   or

Repeated Root (Double Root) of a Quadratic Equation

1. Double Root
   When the two solutions of a quadratic equation are the same, the solution is called a double root.
2. Condition for a Double Root
1. The quadratic equation must be factorable in the form of a perfect square:
   linear expression
2. For to have a double root, must satisfy:

If the coefficient of is not , first divide the equation by the coefficient of to make it before using the above method.

Solving Quadratic Equations Using Square Roots

1. Using Square Roots to Solve a Quadratic Equation
1. For the equation , the solutions are:
2. For the equation , the solutions are:


2. Number of square roots and solutions of quadratic equations:

   Quadratic Equation
   			No solutions
   			No solutions

3. Solving Quadratic Equations Using Completing the Square
   To solve by completing the square:
1. Divide both sides by to make the coefficient of equal to :
   
2. Move the constant term to the right side:
   
3. Add to both sides.
4. Express the left-hand side as a perfect square:
   
5. Solve for using square roots:
   if

Quadratic Formula

1. The solutions of the quadratic equation are given by the quadratic formula:
   if
2. If the coefficient of is an even number, the formula can be simplified:
   if

There are no solutions if or .

Various Types of Quadratic Equations

1. Quadratic Equations with Parentheses
   Use distributive property or multiplication formulas to expand and arrange into the form .
2. Quadratic Equations with Decimal or Fractional Coefficients
   Multiply both sides by an appropriate number to convert the coefficients into integers:
1. If coefficients are decimals, multiply by a power of .
2. If coefficients are fractions, multiply by the least common denominator of the denominators.
3. Quadratic Equations with Common Parts
   Solve in the following steps:
1. Substitute the common part with a variable .
2. Solve for using factoring or the quadratic formula.
3. Substitute back to find the values of .