preview/summary of quadratic equations & functions quadratic – means second power recall...

PREVIEW/SUMMARY OF QUADRATIC EQUATIONS

& FUNCTIONS

QUADRATIC – means second power

Recall LINEAR – means first power

METHOD 1 - FACTORING

Set equal to zero Factor Use the Zero Product Property to solve

(Each factor with a variable in it could be equal to zero.)


Any # of terms – Look for GCF factoring first!

1. 5x2 = 15x5x2 – 15x = 0

5x (x – 3) = 0

5x = 0 OR x – 3 = 0

x = 0 OR x = 3

{0, 3}


Binomials – Look for Difference of Squares

2. x2 = 9x2 – 9 = 0

(x + 3) (x – 3) = 0

x + 3 = 0 OR x – 3 = 0

x = – 3 OR x = 3

{– 3, 3}

Conjugates


Trinomials – Look for PST (Perfect Square Trinomial)

3. x2 – 8x = – 16x2 – 8x + 16 = 0

(x – 4) (x – 4) = 0

x – 4 = 0 OR x – 4 = 0

x = 4 OR x = 4

{4 d.r.}

Double Root


Trinomials – Look for Reverse of Foil

4. 2x3 – 15x = 7x2

2x3 – 7x2 – 15x = 0

(x) (2x2 – 7x – 15) = 0

(x) (2x + 3)(x – 5) = 0

x = 0 OR 2x + 3 = 0 OR x – 5 = 0

{-3/2, 0, 5}

x = 0 OR x = – 3/2 OR x = 5

METHOD 2 – SQUARE ROOTS OF BOTH SIDES

Reorder terms IF needed Works whenever form is (glob)2 = c Take square roots of both sides (Remember

you will need a sign!) Simplify the square root if needed Solve for x. (Isolate it.)


1. x2 = 9

x = 3

Note means both +3 and -3!

x = -3 OR x = 3

{-3, 3}2x = 9


2. x2 = 18

2x = 18

x = 9 2

x = 3 2

or x = 3 2 x = 3 2

OR3 2

-3 2 3 2,


3. x2 = – 9

2x = -9

= -9xCannot take a square root of

a negative. There are NO real number solutions!


4. (x-2)2 = 9

2x - 2 = 9

x -2= 3x = 2 3

This means: x = 2 + 3 and x = 2 – 3x = 5 and x = – 1

{-1, 5}


Rewrite as (glob)2 = c first if necessary.

5. x2 – 10x + 25 = 9

x = 8 and x = 2

{2, 8}(x – 5)2 = 9

2x -5 = 9

x -5= 3x = 5 3



6. x2 – 10x + 25 = 48(x – 5)2 = 48

2x -5 = 16 3

x -5= 4 3

x = 5 4 3

OR5 4 3

5 - 4 3,5+ 4 3

METHOD 3 – COMPLETE THE SQUARE

• Goal is to get into the format: (glob)2 = c• Method always works, but is only

recommended when a = 1 or all the coefficients are divisible by a

• We will practice this method repeatedly and then it will keep getting easier!


Example: 3x2 – 6 = x2 + 12x

2x2 – 12x – 6 = 0Simplify and write in

standard form: ax2 + bx + c = 0

x2 – 6x – 3 = 0 Set a = 1 by division Note: in some problems

a will already be equal to 1.


(x – 3)2 = 12 Rewrite as (glob)2 = c

x2 – 6x – 3 = 0Move constant to other side Leave space to replace it!

x2 – 6x = 3

x2 – 6x + 9 = 3 + 9 Add (b/2)2 to both sides This completes a PST!

METHOD 3 – COMPLETE THESQUARE

Solve for x

Take square roots of both sides – don’t forget

Simplify

(x – 3)2 = 12

x -3 2 3

x 3 2 3

4 3x -3 2x -3 12

OR3 2 3 3 - 2 3 3+2 3,

METHOD 4 – QUADRATICFORMULA

• This is a formula you will need to memorize!• Works to solve all quadratic equations• Rewrite in standard form in order to identify

the values of a, b and c.• Plug a, b & c into the formula and simplify!• QUADRATIC FORMULA:

2-b± b -4acx =2a


Use to solve: 3x2 – 6 = x2 + 12x

2-b± b -4acx =2a

Standard Form: 2x2 – 12x – 6 = 0

2-(-12)± (-12) -4(2)(-6)x =2(2)


12± 144+48 12± 192x =4 4

12± 64 3x =4

12±8 3x =4

3 ± 2 3

2-(-12)± (-12) -4(2)(-6)x =2(2)

a. The graph is a parabola. Opens up if a > 0 and down if a < 0.

b. To find x-intercepts: – may have Zero, One or Two x-intercepts1. Set y or "f(x)" to zero on one side of the equation2. Factor & use the Zero Product Prop to find TWO x-intercepts

c. To find y-intercept, set x = 0. Note f(0) will equal c. I.E. (0, c) d. To find the coordinates of the vertex (turning pt):

1. x-coordinate of the vertex comes from this formula: 2. plug that x-value into the function to find the y-coordinate

e. The axis of symmetry is the vertical line through vertex: x =

REVIEW – QUADRATICFUNCTIONS

b2a

b2a

REVIEW – QUADRATICFUNCTIONS

Example Problem: f(x) = x2 – 2x – 8

a. Opens UP since a = 1 (that is, positive)

b. x-intercepts: 0 = x2 – 2x – 80 = (x – 4)(x + 2)(4, 0) and (– 2, 0)

c. y-intercept: f(0) = (0)2 – 2(0) – 8 (0, – 8)

d. vertex:

e. axis of symmetry: x = 1

-2- , 1, f(1) 1, -92(1)

PRACTICE METHOD 2 – SQUARE ROOTS OF BOTH SIDES

1. x2 = 121

x = 11

Note means both +11 and -11!

x = -11 OR x = 11

{-11, 11}2x = 121

2. x2 = – 81

Cannot take a square root of a negative. There are NO

real number solutions!


2x = -81

= -81x



3. 6x2 = 156

2x 26 =

x2 = 26

x = 26

OR 26 - 26 + 26,

4. (a – 7)2 = 3


2a -7 = 3

a-7= 3a =7 3

OR7 ± 3 7 - 3 , 7 + 3



5. 9(x2 – 14x + 49) = 4

{6⅓, 7⅔}

(x – 7)2 = 4/9

2 4x -7

9 =

OR

2 23 3

2 23 3

x -7 = x =7

x =7 x =7


6. 21 94 64

=0t

21 9t =4 64

9

162t

2 21 9=

4 64

94 t16

t

OR

3 3 3

4 4 4± - ,

PRACTICE METHOD 3 – COMPLETE THE SQUARE

Example: 2b2 = 16b + 6

2b2 – 16b – 6 = 0Simplify and write in


b2 – 8b – 3 = 0 Set a = 1 by division Note: in some problems



(b – 4)2 = 19 Rewrite as (glob)2 = c

b2 – 8b – 3 = 0Move constant to other side Leave space to replace it!

b2 – 8b = 3

b2 – 8b + 16 = 3 +16 Add (b/2)2 to both sides This completes a PST!


Solve for the variable


Simplify

(b – 4)2 = 19

2b - 4 19

19b-4

b 4 19

OR4 19 4 - 19 4+ 19,


Example: 3n2 + 19n + 1 = n - 2

3n2 + 18n + 3 = 0Simplify and write in


n2 + 6n + 1 = 0 Set a = 1 by division Note: in some problems



(n + 3)2 = 8 Rewrite as (glob)2 = c

n2 + 6n + 1 = 0Move constant to other side Leave space to replace it!

n2 + 6n = -1

n2 + 6n + 9 = -1 + 9 Add (b/2)2 to both sides This completes a PST!


Solve for the variable


Simplify

(n + 3)2 = 8

2n+3 8

4 2n+3

n -3 2 2 OR-3 2 2 -3 - 2 2 -3+2 2,


1. x2 – 10x = -3

What number “completes each square”?1. x2 – 10x + 25 = -3 + 25

2. x2 + 14 x = 1 2. x2 + 14 x + 49 = 1 + 49

3. x2 – 1x = 5 3. x2 – 1x + ¼ = 5 + ¼

4. 2x2 – 40x = 4 4. x2 – 20x + 100 = 2 + 100


1. x2 – 10x + 25 = -3 + 25

Now rewrite as (glob)2 = c1. (x – 5)2 = 22

2. x2 + 14 x + 49 = 1 + 49 2. (x + 7)2 = 50

3. x2 – 1x + ¼ = 5 + ¼ 3. (x – ½ )2 = 5 ¼

4. x2 – 20x + 100 = 2 + 100 4. (x – 10)2 = 102


2k - 6 34

k - 6 34

k 6 34

⅓k2 = 4k - ⅔Show all steps to solve.

k2 = 12k - 2

k2 - 12k = - 2

k2 - 12k + 36 = - 2 + 36

(k - 6)2 = 34

6 34 6 34,

PRACTICE METHOD 4 – QUADRATIC FORMULA

1 1 + 48

4

1 49

4

2x2 = x + 6Show all steps to solve & simplify.

2x2 – x – 6 =0

1 1 - 4(2)(-6)

2(2)

and

3- , 22

1 7 -6 8

4 4 4


1 1 - 4(1)(5)

(2)(1)

x2 + x + 5 = 0

Show all steps to solve & simplify.

1 1- 20 1 -19

2 2

not a real number


-2 4 - 4(1)(-4)

(2)(1)

-2 4+16 -2 20

2 2

2 -1 5-2 4 5 -2 2 5

2 2 2(1)

x2 +2x - 4 = 0

Show all steps to solve & simplify.

-1 5

THE DISCRIMINANT – MAKING PREDICTIONS

Four cases:b2 – 4ac is called the discriminant

1. b2 – 4ac positive non-square two irrational roots

2. b2 – 4ac positive square two rational roots

3. b2 – 4ac zero one rational double root

4. b2 – 4ac negative no real roots


Use the discriminant to predict how many “roots” each equation will have.

1. x2 – 7x – 2 = 0

2. 0 = 2x2– 3x + 1

3. 0 = 5x2 – 2x + 3

4. x2 – 10x + 25=0

49–4(1)(-2)=57 2 irrational roots

9–4(2)(1)=1 2 rational roots

4–4(5)(3)=-56 no real roots

100–4(1)(25)=0 1 rational double root

THE DISCRIMINANT – MAKING PREDICTIONS about Parabolas

The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located.

1. y = 2x2 – x - 6

2. f(x) = 2x2 – x + 6

3. y = -2x2– 9x + 6

4. f(x) = x2 – 6x + 9

1–4(2)(-6)=49 2 rational zerosopens up/vertex below x-axis/2 x-intercepts

1–4(2)(6)=-47 no real zerosopens up/vertex above x-axis/No x-intercepts

81–4(-2)(6)=129 2 irrational zerosopens down/vertex above x-axis/2 x-intercepts

36–4(1)(9)=0 one rational zeroopens up/vertex ON the x-axis/1 x-intercept


Note the proper terminology:

The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have.

The “roots” of an equation are the x values that make the expression equal to zero.

Equations have roots. Functions have zeros which are the x-intercepts on it’s graph.

FOUR METHODS – HOW DO I CHOOSE?

Some suggestions:

Quadratic Formula – works for all quadratic equations, but first look for a “quicker” method. You must rewrite into standard form before using Quad Formula! Don’t forget to simplify square roots and use value of the discriminant to predict the number of roots or zeros.

Square Roots of Both Sides – use when the problem can easily be put into the form: glob2 = constant.

Examples: 3(x + 2)2=12 or x2 – 75 = 0

FOUR METHODS – HOW DO I CHOOSE?

Some suggestions:

Factoring – doesn’t always work, but IF you can see the factors, this is probably the quickest method.

Examples: x2 – 8x = 0 has a GCF4x2 – 12x + 9 = 0 is a PSTx2 – x – 6 = 0 is easy to FOIL

Complete the Square – It always works, but if you aren’t quick at arithmetic with fractions, then this method is best used when a = 1 and b is even (so no fractions).

Example: x2 – 6x + 1 = 0

QUADRATIC FORMULA – Derive it by Completing the Square!

2x + xb

a+

c

a= 0

2 bx + x + =

a

c-a

22 2

2 2

b b+ +

b cx +

4a 4ax = -

a a

Start with Standard Form:ax2 + bx + c = 0

2

2

2b c

x = -b 4a•

+2 4aa 4a•a

2

22b - 4ab

x + =2

c

a 4a

2bbx +

- 4

2a

ac= ±

2a2b - 4ac

x = - ±b

2a 2a

2

2

2b b - 4a

±c

x + =2a 4a

2-b ± b - 4acx =

2a

preview/summary of quadratic equations & functions quadratic – means second power recall...

Documents