Solving Quadratic Equations: Factoring

Solving Quadratic Equations: Factoring

• Definition of Quadratic Equations:

  • A quadratic equation is a polynomial of the form ( ax^2 + bx + c = 0 ) where ( a \neq 0 ).

• Factoring Method:

  • Used to rewrite the quadratic equation in the form ( (px + q)(rx + s) = 0 ).

• Steps to Factor a Quadratic Equation:

  1. Identify Coefficients:

     • Determine ( a ), ( b ), and ( c ) from the equation ( ax^2 + bx + c = 0 ).

  2. Find Two Numbers:

     • Look for two numbers that multiply to ( ac ) (the product of ( a ) and ( c )) and add to ( b ).

  3. Rewrite the Equation:

     • Split the middle term ( bx ) into two terms using the two numbers found.
     • For example, if the numbers are ( m ) and ( n ), rewrite ( bx ) as ( mx + nx ).

  4. Factor by Grouping:

     • Group the terms into two pairs and factor out the common factors from each pair.
     • Example: From ( ax^2 + mx + nx + c ), group as ( (ax^2 + mx) + (nx + c) ).

  5. Set Each Factor to Zero:

     • Once factored, set each factor equal to zero to solve for ( x ):
       • ( px + q = 0 )
       • ( rx + s = 0 )

• Example:

  • Given ( x^2 + 5x + 6 = 0 ):
    1. Identify ( a = 1 ), ( b = 5 ), ( c = 6 ).
    2. Find numbers that multiply to ( 6 ) (1, 6) and add to ( 5 ) (2, 3).
    3. Rewrite as ( x^2 + 2x + 3x + 6 = 0 ).
    4. Factor: ( (x + 2)(x + 3) = 0 ).
    5. Solve: ( x + 2 = 0 ) or ( x + 3 = 0 ) gives ( x = -2 ) or ( x = -3 ).

• When Factoring is Not Possible:

  • If the quadratic does not factor cleanly, use methods like completing the square or the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

• Special Cases:

  • Perfect Square Trinomials: ( (x + p)^2 = x^2 + 2px + p^2 ).
  • Difference of Squares: ( a^2 - b^2 = (a - b)(a + b) ).

• Checking Your Work:

  • Use the FOIL method to ensure that the factors multiply back to the original quadratic.

Quadratic Equations Overview

• A quadratic equation is represented as ( ax^2 + bx + c = 0 ) where ( a \neq 0 ).

Factoring Method

• The goal is to express the quadratic in the factored form ( (px + q)(rx + s) = 0 ).

Steps to Factor a Quadratic Equation

• Identify Coefficients: Extract values of ( a ), ( b ), and ( c ) from the equation.
• Find Two Numbers: Look for two numbers that multiply to ( ac ) and sum to ( b ).
• Rewrite the Equation: Split the term ( bx ) into two parts using the identified numbers.
• Factor by Grouping: Group terms into pairs and factor out common factors from each pair.
• Set Each Factor to Zero: Solve each resulting linear equation by setting ( px + q = 0 ) and ( rx + s = 0 ).

Example of Factoring

• In the equation ( x^2 + 5x + 6 = 0 ):
  • Coefficients identified: ( a = 1 ), ( b = 5 ), ( c = 6 ).
  • Numbers found: 2 and 3, which multiply to 6 and add to 5.
  • The equation is rewritten as ( x^2 + 2x + 3x + 6 = 0 ).
  • Factored as ( (x + 2)(x + 3) = 0 ).
  • Solutions found are ( x = -2 ) and ( x = -3 ).

When Factoring is Not Possible

• If the equation does not factor neatly, alternative methods like completing the square or using the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) should be employed.

Special Cases

• Perfect Square Trinomials: Take the form ( (x + p)^2 = x^2 + 2px + p^2 ).
• Difference of Squares: Expressed as ( a^2 - b^2 = (a - b)(a + b) ).

Checking Your Work

• Employ the FOIL method to verify that the factors multiply correctly back to the original quadratic equation.

This quiz explores the method of factoring quadratic equations. Learn how to identify coefficients, find necessary numbers, and rewrite equations effectively. Test your knowledge with practical examples and enhance your understanding of quadratic equations.