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.

This quiz is designed to test your understanding of solving quadratic equations by factoring. You will explore various true equations and their factored forms, as well as find zeros and solutions. Perfect for reinforcing key concepts in quadratic functions.