Solving Quadratic Equations – Comparing and Contrasting Methods
April 21, 2015
After looking at your practice quizzes, here are the biggest misconceptions that I noticed:
• How do you factor when the leading coefficient (a) is negative?
• How do you determine the solutions to a quadratic equation from a graph?
• What are the differences between “factoring completely” and solving a quadratic equation?
• Remembering when taking square root of both sides of equation
• Writing quadratic formula and extending whole fraction bar underneath numerator
• Explaining differences between quadratic functions, quadratic equations, and the quadratic formula
• What do you do when one method doesn’t work?
There are 5 methods that you have learned for solving quadratic equations:
• Factoring• Completing the Square• Graphing• Taking the Square Root• The Quadratic Formula
Factoring
a = 1 Factor normally
or use British Method
a 1 Factor using
British Method
Solving Quadratic Equations: Factoring
• Solve the quadratic equation by factoring
Solving Quadratic Equations: Factoring Solve the equation by factoring:
Pros: Efficient and easyCons: It doesn’t always work if the roots are irrational
Completing the Square
a = 1 Complete the Square using
normal process
a 1 Divide all terms by a and then complete the
square
Solving Quadratic Equations: Completing the Square
• Solve the quadratic equation by completing the square:
Pros: It always works. Useful in upper level classesCons: Time-consuming and a lot of steps
Solving Quadratic Equations: Completing the Square (
• Solve the quadratic equation by completing the square:
Solving Quadratic Equations: Graphing
• Solve the quadratic equation by graphing with the TI-nSpire or TI-84 graphing calculator:
Pros: Quick, visual, easyCons: You won’t always have a graphing calculator. Difficult to estimate specific roots if roots are irrational
Solving Quadratic Equations:The Quadratic Formula
Solve the quadratic equation by using the quadratic formula:
Pros: It always worksCons: Time-consuming
Solving Quadratic Equations: Taking the Square Root
• Solve the quadratic equation by taking the square root
Pros: Quick and efficientCons: It only works when b = 0
It’s often going to be up to you to solve to choose a method for solving quadratic equations, especially on a college admissions test.
• What’s your plan if you try one method and it doesn’t work?
• How can you know for sure how many solutions a quadratic equation has?
• What, in your opinion, is the best method for solving quadratic equations?
Explain the difference between a quadratic equation, quadratic function, and the quadratic formula.
Up next this week and next week:• Today: Factoring by Completing the Square (cont’d)
and Deriving the Quadratic Formula• Tomorrow: Factoring by Guess-and-Check and
Review Difference of Squares/Perfect Square Trinomials
• Thursday: Factoring Word Problems• Friday: Factoring by Grouping and Distributive
Property and More Word Problems• Monday: Rotated Out• Tuesday: Review Day• Wednesday: Unit Test – Factoring, Solving Quadratic
Equations, and Polynomials
Team Challenge: Complete the Square on the General Case
For this group challenge, you and your team will attempt to complete the square on the general case of a quadratic equation
The specifics:• The first 4 minutes are independent work time so that
you can gather ideas, brainstorm a strategy, and prepare to work with your group.
• You will then have 7 minutes to complete the square with your group
The Prize:• Any group demonstrating their complete process and
correctly completing the square will earn a homework pass for tonight , Wednesday, April 23rd
Ready…Go!
Complete the Square on
Class Discussion: How do you complete the square on the general case
𝑎𝑥2+𝑏𝑥+𝑐=0
Factoring by (Grouping)April 22, 2015
This topic should be familiar – you have already factored by grouping with the British Method!
*The basic idea is to group terms with “common factors” before factoring
Example 1: Factor
Example 2: Factor
Example 3: Factor
Sum and Difference of Cubes Patterns
• Two other special factoring patterns!
Helpful memory trick:
Example 4:
Example 5: Factor
Example 6: Factor
Small-Group Work: Please complete today’s worksheet with your group.