Quadratic Equations

C.A.1-3

Solving Quadratic Equations

• If a quadratic equation ax²+bx+c=0 can’t be solved by factoring that means the solutions involve roots.

• The quadratic formula can be used to solve any quadratic equation by using the coefficients a, b, and c.

02 cbxaxa

cabbx

2

42

*Since the Quadratic Formula involves the square root we can have a variety of different solutions; from rational to radical to complex.

The derivation of the Quadratic Formula

Using the quadratic formula to solve equations….

• Make sure your equation is in the form, ax²+bx+c=0, if it isn’t then use algebra to put it in this form.

• Identify the coefficients a, b, and c.

• Plug them into the equation and simplify the result.

0352 xx3,5,1 cba

2

135

2

12255

12

31455 2

x

2

135,

2

135

These are the two solutions, they are radical solutions

Examples… 53 xx

053

5553

53

2

2

2

xx

xx

xxSolve:

5,3,1 cba

2

293

2

2093

12

51433 2

x

Examples…Solve: xx 10232

02310

10232

2

xx

xx

252

22

2

10

2

810

2

9210010

12

231410)10( 2

x

23,10,1 cba

*The solutions are approximately 6.4142 and 3.5858 if rounded to 4 decimal places.

Solving Quadratic Inequalities• We can use our factoring and

solution methods to determine when a quadratic has positive and negative output values.

• First find the zeros for the quadratic.

• Place them on a number line and test values on each side of the zeros to determine the sign of the region.

• + means the region is positive• - means the regions is

negative• List all of the regions that

satisfy the inequality in interval notation.

012 xx

includednot

xzeros 1,2:

Try x=-2

+Try x=0

-Try x=3

+-1 2

Solution: (-∞,-1)U(2,∞)

Examples….

Solve the inequality:

01032 xx

025 xx

includednot

xzeros 2,5:

Try x=-3

+Try x=0

-Try x=6

+

-2 5

Solution: (-2,5)

Examples….

Solve the inequality:

0562 xx

051 xx

included

xzeros 1,5:

Try x=-6

+Try x=-2

-Try x=0

+

-5 -1

Solution: (-∞,-5]U[-1,∞)

Inverses-3-7 topic

p.305-313

Finding an inverse for a function…

• For a function f(x)=rule, put in y=rule form.• Swap y with x and vice versa.• Solve for y.• The inverse is the function y=new rule and is

denoted:

rulenewxf 1

Examples….

1. Find the inverse for f(x)=2x+3.

2

32

3

23

32

32

32

1

xxf

yx

yx

yx

xy

xxf

Swap the x’s and y’s!

Solve for y!

Examples….

2. Find the inverse of

53

53

533

53

53

5

1

xxf

yx

yx

yx

xy

xxf

3

5x

xf

Swap the x’s and y’s!

Solve for y!