# 1 # 2 # 3 multiply # 4 multiply: multiply # 5 # 6 solve

Warm UpWarm UpWarm Up

# 1# 1# 1# 1 # 2# 2# 2# 2

2x 7# 3# 3# 3# 3

Multiply

# 4# 4# 4# 4

Multiply:

2x 5Multiply:

2x 4

Multiply

# 5# 5# 5# 5 # 6# 6# 6# 6

2x 12

Solve Solve

24x 13 5 2x 12 37


# 1# 1# 1# 1 Multiply: 2x 5

x 5 x 5 2 10x x 25 2 10x x 25

x 5

x

5 25

2x 5x

5x


# 2# 2# 2# 2 Multiply 2x 4 2 8x x 16 2 8x x 16

x 4

x

4 16

2x 4x

4x


# 3# 3# 3# 3 Multiply: 2 14x x 49 2 14x x 49 2x 7


# 4# 4# 4# 4 Multiply 2x 12 2 24x x 144 2 24x x 144


# 5# 5# 5# 5 Solve 2x 12 37

2x 25

2x 25

x 5x 5


# 6# 6# 6# 6 Solve 24x 13 5

24x 8

2x 2

2x 2

x 2x 2


2x 5 2 10x x 252 10x x 25

2x 4 2x 8x 162x 8x 16

2 14x x 49 2 14x x 49 2x 7

2x 12 2x x24 144 2x x24 144

Pg. 585, #17 – 34. Format: Free

Quadratic Equations and FunctionsQuadratic Equations and FunctionsCh 9Ch 9

Quadratic Equations and FunctionsQuadratic Equations and FunctionsCh 9Ch 9

Using Using Square RootsSquare Roots to Solve Quadratic Equations to Solve Quadratic EquationsUsing Using Square RootsSquare Roots to Solve Quadratic Equations to Solve Quadratic Equations

Solving equations in the form: 2x k c

2x xa b c 0

Exact solutions

Approx. solutions

x 7

x 2.65

x 4 13

x 7.6 and 0.4

Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations

Example 1Example 1Example 1Example 1

2x k c 2x k c

Solve: 2x 4 64 2(x 4) 64

x 4 8

x 4 x 12

x 4 8


Example 2Example 2Example 2Example 2

2x 7 23 Solve:

x 7 23

x 7 23

2(x 7) 23

x 7 23 x 7 23


Example 3Example 3Example 3Example 3 Solve

2(x 5) 81

x 5 81

x 5 81

x 5 9

x 14x 14 x 4x 4



2(x 12) 47

x 12 47

x 12 47

x 12 47 x 12 47 x 12 47 x 12 47



2(3x 2) 49

3x 2 7

3x 2 7

x 3x 3

5x

3

5x

3

3x 2 7 3x 2 73x 9 3x 5



2(x 1) 16

x 1 16



2(x g) h

x g h

x g h

x g h x g h x g h x g h

Deriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic Formula

2x xa b c 0

2a b x x c

2 b c x

ax

a

2b

x2a

2 2

2 2

b b 4ax

2a 4a 4c

a

2 2

2

b b 4acx

2a 4a

2

2

b b 4acx

2a 4a

2b b 4acx

2a 2a

2b b 4acx

2a 2a

2b b 4acx

2a

2b b 4ac

x2a

2x xa b c y

2

2

b4a

ca

2 2

2 2

b b

4a 4a

Deriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic Formula

2 2

2

b b 4acx

2a 4a

2

2

b b 4acx

2a 4a

2b b 4acx

2a

2b b 4ac

x2a

2x xa b c y

Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems

2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6

2(x 3) 21 # 3# 3# 3# 3# 4# 4# 4# 4 2(x 4) 36

2(x 5) 10 # 5# 5# 5# 5 # 6# 6# 6# 6 2(x 9) 144

2(2x 1) 81 # 7# 7# 7# 7 # 8# 8# 8# 8 2(x 5) 9

Solve using square roots

2(x 3) 16


2(x 3) 16

x 3 4

Take the square root of both sides of the equation(this is the inverse operation that “un-does” the exponent)

Remember to include both possibilities for the square root,i.e. the positive root and the negative root.

You now have essentially two linear equations and you’ll need to solve each to find the solutions to the original quadratic equation.

x 3 4 x 7

x 3 4 x 1

Example #1Example #1Example #1Example #1

Notice that the number 16 is a perfect square, so taking the square root of it results in a rational number

2(x 5) 13

2(x 5) 13

x 5 13

Just like example #1, you take the square root of both sides of the equation to get rid of the exponent.

Remember to include both possibilities for the square root, i.e. the positive root and the negative root (do not approximate for tonight’s homework – in other words, do not use a calculator to find the decimal representation of the irrational number.)

You now have essentially two linear equations and you’ll need to solve each to find the solutions to the original quadratic equation.

x 5 13

x 5 13

Example #2Example #2Example #2Example #2

Notice that the number 13 is NOT a perfect square, so taking the square root of it results in an irrational number.

x 5 13

x 5 13

Note: it is customary to express the solution like this: x 5 13


2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6

2(x 3) 21 # 3# 3# 3# 3# 4# 4# 4# 4 2(x 4) 36

2(x 5) 10 # 5# 5# 5# 5 # 6# 6# 6# 6 2(x 9) 144

2(2x 1) 81 # 7# 7# 7# 7 # 8# 8# 8# 8 2(x 5) 9



2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6


2(x 13) 64

x 13 8

x 21 x 5

x 13 8

2(x 3) 6

x 3 6

x 3 6 x 3 6

x 3 6


# 3# 3# 3# 3# 4# 4# 4# 42(x 3) 21 2(x 4) 36


2(x 3) 21

x 3 21

x 3 21 x 3 21

x 3 21

2(x 4) 36

x 4 6

x 2 x 10

x 4 6


# 5# 5# 5# 5# 6# 6# 6# 62(x 5) 10 2(x 9) 144


2(x 5) 10

x 5 10

x 5 10 x 5 10

x 5 10

2(x 9) 144

x 9 12

x 21 x 3

x 9 12


# 7# 7# 7# 7# 8# 8# 8# 82(2x 1) 81 2(x 5) 9


2(2x 1) 81

2x 1 9

x 5 x 4

2x 1 9

2(x 5) 9

2x 10 2x 8

# 1 # 2 # 3 multiply # 4 multiply: multiply # 5 # 6 solve

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quadratic equations

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positive root

original quadratic equation