CHAPTER 5

• EXPRESSIONS AND FUNCTIONS• GRAPHING• FACTORING• SOLVING BY:

– GRAPHING– FACTORING– SQUARE ROOTS– COMPLETING THE SQUARE– QUADRATIC FORMULA

General Form General Form of a of a

Quadratic EquationQuadratic Equation

y = ax2 + bx + c

Axis of Symmetry

This is the equation of a vertical line.The vertex will be located on this line.

Vertex

• The vertex is a point of maximum or minimum. Each side of the parabola is symmetric about this point.

• The x-coordinate of the vertex is

• The y-coordinate of the vertex is

a

b

2

a

bf

2

Other Points

• Choose a few x values on either side of the axis of symmetry.

• Calculate the value of the function at those locations, then graph the corresponding points on the other side of the axis of symmetry.

Example

762 xxy

Factoring Quadratic Expressions

cbxax 2

First note that this is an expression; it does not equal anything.

Here is the process in words: List the factors of a and thefactors of c. These will be the possible firsts and lasts in the two binomial factors. Since they are also the outers and the inners, you must check to see which pairs will have the sum b. If the expression is factorable, there will only be one way in which the factors will work with appropriate signs.

Factoring Example

Special Factoring Situations

• Perfect Squared Trinomials

• Differences of Squares

Perfect Squared Trinomials

“The first term squared plus twice the first times the last plus the last term squared.”

22 2 babax

Since the middle term is “twice the first times the last”, it means that the outer and inner terms of the binomial factors must be the same. In other words, (a+b)(a+b).

2ba Go back

Difference of Squares

baba

22 ba This can be thought of as ,where m = 0. This really means that the outer and inner terms must be opposite. So the factored formis obvious:

22)( bma

Simplifying Radicals

• No perfect squared factors

• No fractions under the radical

• No radicals in the denominator

Solving Quadratic Equations

• Graphing

• Square Roots

• Factoring

• Completing the Square

• Quadratic Formula

Graphing

0762 xx

To solve by graphing, replace thezero with “y”. The solutions willbe the points on the graph wherey = 0.

Note that this method works well whenthe zeros are easily within a normal window,and when factoring is not possible. If the answers arenot rational, then the calculator will only give approximations.This could be a drawback for this method. Go back

Square Roots

• Use this method when there is not a linear term or when there is a perfect square that can be isolated.

• Also remember that taking the square root of both sides of the equation is the last step.

• Of most importance, is the fact that there will be two possible solutions, one positive and one negative.

Example

362 x

6x

362 x

Another Example

495 2 x 49)5( 2 x

75 x75 x 75 x2x 12x

Go back

Factoring

The principle here is that once factored, the product is zero.This means that either one or the other of the factors is zero.So set each variable expression equal to zero and find the valueof the variable that makes the equation true.

IF ab = 0, THEN EITHER

a = 0 OR b = 0.

Example

Go back

Completing the Square

• This method takes any trinomial and turns it into a problem like .

• The method is as follows: – make sure a = 1– make sure only the quadratic and linear terms

are one a side of the equation– find half of the coefficient of the linear term,

square it and add it to both sides of the equation.

495 2 x

Further steps

• Now the side of the equation with the variables is a perfect square.

• Factor it and use square roots to solve.

Example

Go back

The Discriminant

• The discriminant will tell you how many real solutions the equation has.

• None, 1, or 2

• It will also tell you if the trinomial is a perfect square or not.

acb 42

Quadratic Formula

• This method always works.

• This method requires that the equation be in the form of ax2 + bx + c = 0.

• This method should be used when factoring is not obvious or when exact solutions are needed.

The FORMULA!

a

acbbX

2

42

Example

)1(2

)7)(1(466 2 x

0762 xx

72

86

12

86

2

86

2

646

2

28366

x

x

x

x

x