Deriving the Quadratic Formula

Author: Jessica Brockway

Length of Lesson: 45 minutes

Class: Algebra 1

Source(s) of Lesson:

http://www.themathpage.com/alg/complete-the-square.htm#proof

TEKS addressed:

(A.10.B) make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.

(A.4.A) find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations

(A.4.B) use the commutative, associative, and distributive properties to simplify algebraic expressions

Learner Outcomes: Students will be able to solve quadratic equations using the quadratic formula. They will also know how to derive the quadratic formula from the standard form of a quadratic equation.

Supplementary materials:

Graphing calculators

Attached worksheet

TEACHER DOES	STUDENT DOES
ENGAGEMENT
Have students work on problem when they arrive in class: The area of a rectangle is given by the equation 2l^2 - 5l = 18, in which l is the rectangle’s length. What is the length of the rectangle? F 1.5 G 2 H* 4.5 J 6 Let students explain to class how to solve.	Have students solve problem using factoring
EXPLORATION
This week we have learned two ways of solving quadratic equations. Who can tell me what they are? Correct. There is also another way to solve a quadratic equation, called the Quadratic Formula. (writes it on the board) x = [ -b ± √(b^2 - 4ac) ] / 2a what do the letters a, b, and c correspond to? Correct Lets do an example: Solve x^2 + 3x – 4 = 0 using the quadratic equation (answer: x=-4, x=1) Now what do the x’s represent? What are the roots? Correct. Now do you think this formula will work for all quadratic equations?	Factoring and completing the square The quadratic equation: ax^2+ bx + c = 0 Students work alone or in partners to solve. The roots. The roots are where the parabola (line) cross the x axis. Yes/No/I don’t know
EXPLANATION
How do we know that the quadratic formula is correct and that it will always give us the correct roots? We must give a proof. Can we turn the equation ax^2+ bx + c = 0 into x = [ -b ± √(b^2 - 4ac) ] / 2a? Lets assume that we can. What would be the first step? How would we isolate x? x^2+(b/a)x+(c/a)=0 why is it still zero at the end? What do we do next? x^2+(b/a)x= -(c/a) now what do we do? What did you learn last class to solve quadratic equations? x^2+(b/a)x+(b^2/4a^2) = (b^2/4a^2) – (c/a) Now what can we do? (x+b/2a)^2= (b^2/4a^2) – (4ac/4a^2) (x+b/2a)^2=(b^2-4ac)/ 4a^2 x+b/2a= ± √[(b^2 - 4ac)/ 4a^2] why is it ± ? What is it that we are trying to do again? x+b/2a= ± √(b^2 - 4ac)/2a x= (-b/2a) ± √(b^2 - 4ac)/2a x = [ -b ± √(b^2 - 4ac) ] / 2a	Plug it in the calculator. You have to write a proof. Yes/no/students are unsure Want x all by itself (isolate x) Divide by a Zero divided by anything is still zero Complete the square Rewrite the left side Make the right side have common denominators Now combine the terms on the right Take the square root of each side Because it is a square root – both have the same answer Get x by itself (isolated) They have the same denominator - you can combine them.
EVALUTION
Give students attached worksheet	Students work on worksheet at the end of class and finish for homework if needed.
ELABORATION
(if there is enough time/students finish worksheet) Have the students get in groups and work on the following problem: 5x^2– 2x + 2 = 0 where are you getting stuck? How is this problem different from the ones on the worksheet? You can take the square root of a negative number, but it is an imaginary number, not a rational number.	Students start working on the problem We are getting stuck on the square root How do we take the square root of a negative number? Can we take the square root of a negative number?

WORKSHEET

Use the quadratic formula to find the roots of each quadratic:

1.) x² - 5x + 5 = 0

2.) 2x² – 8x +5 = 0

3.) x² + 6x +2 = 0