Algebra: Solving Quadratics by Factoring

Lesson 1: Solving Quadratics by Factoring

• Quadratic equations can be solved by factoring into products of binomials.
• True equations include:
  • ( 9x^2 - 1 = (3x - 1)(3x + 1) )
  • ( x^2 - 169 = (x - 13)(x + 13) )
  • ( 16x^2 - 64y^2 = (4x - 8y)(4x + 8y) )

Factored Forms of Quadratics

• Factored form of ( x^2 - 22x + 121 ) is ( (x - 11)^2 ).
• Factored form of ( 2m^3 - 26m^2 + 80m ) is ( 2m(m - 5)(m - 8) ).
• Factored form of ( x^2 - 8x - 48 ) is ( (x + 4)(x - 12) ).
• Factored form of ( x^2 - 2x - 63 ) is ( (x + 7)(x - 9) ).
• Factored form of ( 3x^2 + 6x - 24 ) is ( 3(x + 4)(x - 2) ).
• Factored form of ( 9x^2 + 42x + 49 ) is ( (3x + 7)^2 ).
• Factored form of ( x^2 - 7x - 18 ) is ( (x + 2)(x - 9) ).
• Factored form of ( 36x^2 - 49y^2 ) is ( (6x + 7y)(6x - 7y) ).

Zeros and Solutions

• Zeros of ( f(x) = x^2 - 10x + 21 ) are ( 3 ) and ( 7 ).
• Solutions to ( 3(x + 2)(x - 9) = 0 ) are ( -2 ) and ( 9 ).
• Solutions to the equation ( 40 = 2x^2 + 2x ) are ( -5 ) and ( 4 ).
• Zeros of ( f(x) = x^2 - 36 ) occur at ( x = 6 ) and ( x = -6 ).
• Solutions to ( (x - 4)(x + 2) = 0 ) are ( -2 ) and ( 4 ).
• Solutions to ( x^2 + 21x + 110 = 0 ) are ( -10 ) and ( -11 ).
• Solutions of ( 25b^2 - 64 = 0 ) are ( \frac{8}{5} ) and ( -\frac{8}{5} ).

Common Mistakes

• Mistakes can occur in applying factoring patterns correctly, such as the difference of squares.
• Ensure to consider both positive and negative roots when solving equations.
• Be cautious about dividing all parts of an equation equally when simplifying.

Lesson 2: Other Methods for Solving Quadratics

• Squaring both sides can yield two solutions, such as ( (x - 9)^2 = 25 ) yielding ( 14 ) and ( 4 ).
• Some equations may have no real solutions, for instance, ( x^2 + 38 = 16 ).

Completing the Square

• Completing the square can provide solutions in a different form, such as:

  • For ( -4x^2 + 6x + 16 = 0 ), solutions are ( 3 \pm \frac{\sqrt{73}}{4} ).
  • For ( -x^2 + 8x - 4 = 0 ), solutions are ( 4 \pm 2\sqrt{3} ).

• Notable mistakes in completing the square include not handling constants correctly or neglecting the negative square root.

Summary of Solutions

• Common solutions to equations include ( 8, -8 ) for ( x^2 + 15 = 79 ) and ( -\frac{5}{6}, \frac{5}{6} ) for ( x^2 = \frac{25}{36} ).
• The equation ( (x + 49)^2 = 24 ) simplifies to ( x = -49 \pm 2\sqrt{6} ).

Overall Strategies

• Familiarize with different methods such as factoring, completing the square, and recognizing when an equation has no real solutions.
• Practice deriving zeros, factoring, and correcting mistakes in approaches to enhance problem-solving skills.