Intermediate AlgebraChapter 11

•Quadratic Equations

Willa Cather –U.S. novelist

• “Art, it seems to me, should simplify. That indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole – so that all one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.

Intermediate Algebra 11.1

•Special Methods

Def: Quadratic Function

• General Form• a,b,c,are real numbers and a not equal 0

2( )f x ax bx c

Solving Quadratic Equation #1

• Factoring• Use zero Factor Theorem• Set = to 0 and factor• Set each factor equal to zero• Solve• Check

Solving Quadratic Equation #2

• Graphing• Solve for y• Graph and look for x intercepts• Can not give exact answers• Can not do complex roots.

Solving Quadratic Equations #3Square Root Property

• For any real number c2if x c then

x c or x c

x c

Sample problem2 40x

40x 4 10x

2 10x

Sample problem 225 2 62x

25 60x 2 12x

12x 2 3x

Solve quadratics in the form

2ax b c

Procedure

• 1. Use LCD and remove fractions• 2. Isolate the squared term• 3. Use the square root property• 4. Determine two roots• 5. Simplify if needed

Sample problem 3

23 16x 3 16x 3 4x

3 4 3 4 3 4x x or x

1 7 1, 7x or x

Sample problem 4

27 25 2 3 0x 225 2 3 7x 2 72 3

25x

7 72 325 5

x i

3 7 1.5 0.262 10

x i i

Dorothy Broude

•“Act as if it were impossible to fail.”

Intermediate Algebra 11.1 Gay

•Completing • the

•Square

Completing the square informal

• Make one side of the equation a perfect square and the other side a constant.

• Then solve by methods previously used.

Procedure: Completing the Square• 1. If necessary, divide so leading

coefficient of squared variable is 1.• 2. Write equation in form• 3. Complete the square by adding the

square of half of the linear coefficient to both sides.

• 4. Use square root property• 5. Simplify

2x bx k

Sample Problem

2 8 5 0x x

4 11x

Sample Problem complete the square 2

2 5 1 0x x 5 29

2x

Sample problem complete the square #3

23 7 10 4x x

7 236 6

x i

Objective:

• Solve quadratic equations using the technique of completing the square.

Mary Kay Ash

• “Aerodynamically, the bumble bee shouldn’t be able to fly, but the bumble bee doesn’t know it so it goes flying anyway.”

Intermediate Algebra 11.2

•The •Quadratic •Formula

Objective of “A” students

• Derive • the

• Quadratic Formula.

Quadratic Formula

• For all a,b, and c that are real numbers and a is not equal to zero

23 8 7 0

4 53 3

x x

x i

2 42

b b acxa

Sample problem quadratic formula #1

22 9 5 0x x 1 , 52

Sample problem quadratic formula #2

2 12 4 0x x

6 2 10x

Sample problem quadratic formula #3

23 8 7 0x x 4 5

3 3x i

Pearl S. Buck

• “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”

Methods for solving quadratic equations.

• 1. Factoring• 2. Square Root Principle• 3. Completing the Square• 4. Quadratic Formula

Discriminant

• Negative – complex conjugates• Zero – one rational solution (double

root)• Positive

– Perfect square – 2 rational solutions– Not perfect square – 2 irrational

solutions

2 4b ac

Sum of Roots

1 2br ra

Product of Roots

1 2cr ra

CalculatorPrograms

• ALGEBRAQUADRATIC• QUADB• ALG2• QUADRATIC

Harry Truman – American President

• “A pessimist is one who makes difficulties of his opportunities and an optimist is one who makes opportunities of his difficulties.”

Intermediate Algebra 11.4

•Quadratic Inequalities

Sample Problem quadratic inequalities #1

2 2 8 0x x 2,4

Sample Problem quadric inequalities #2

26 2x x 1 2, ,

2 3

Sample Problem quadratic inequalities #3

2 6 9 0x x ,R

Sample Problem quadratic inequalities #4

4 01

xx

( 1,4]

Sample Problem quadratic inequalities #5

3 2 02 3x x

2,3 5,

Intermediate Algebra 11.5-11.6

•Quadratic Functions

Orison Swett Marden

• “All who have accomplished great things have had a great aim, have fixed their gaze on a goal which was high, one which sometimes seemed impossible.”

Vertex

• The point on a parabola that represents the absolute minimum or absolute maximum – otherwise known as the turning point.

• y coordinate determines the range.• (x,y)

Axis of symmetry

• The vertical line that goes through the vertex of the parabola.

• Equation is x = constant

Objective

• Graph, determine domain, range, y intercept, x intercept

2

2

y x

y ax

Parabola with vertex (h,k)

• Standard Form

2y a x h k

Find Vertex

• x coordinate is

• y coordinate is

2ba

2bfa

Graphing Quadratic• 1. Determine if opens up or down• 2. Determine vertex• 3. Determine equation of axis of symmetry• 4. Determine y intercept• 5. Determine point symmetric to y

intercept• 6. Determine x intercepts• 7. Graph

Sample Problems - graph

2 6 5y x x 2 2 3y x x 23 6 1y x x

Roger Maris, New York Yankees Outfielder

•“You hit home runs not by chance but by preparation.”