PREVIEW/SUMMARY OF QUADRATIC EQUATIONS

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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• QUADRATIC means second power
• Recall LINEAR means first power

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Click on the Quadratic Method you wish to review
to go to those slides

• Quadratic Formula
• Always works, but be sure to rewrite in standard
  form first of ax2 bx c 0
• How to Choose Among
• Methods 1 4?
• Summary Hints
• Quadratic Functions Summary of graphing
  parabolas from
• f(x) ax2 bx c form

Factoring Method Quick, but only works for
some quadratic problems Square Roots of Both
Sides Easy, but only works when you can get in
the form of (glob)2 constant Complete the
Square Always works, but is recommended only
when a 1 or all terms are evenly divisible to
set a to 1.
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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.)

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METHOD 1 - FACTORING

• Any of terms Look for GCF factoring first!
• 5x2 15x

5x2 15x 0
0, 3
5x (x 3) 0
5x 0 OR x 3 0
x 0 OR x 3
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METHOD 1 - FACTORING

• Binomials Look for Difference of Squares
• 2. x2 9

x2 9 0
3, 3
(x 3) (x 3) 0
x 3 0 OR x 3 0
x 3 OR x 3
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METHOD 1 - FACTORING

• Trinomials Look for PST (Perfect Square
  Trinomial)
• 3. x2 8x 16

x2 8x 16 0
4 d.r.
(x 4) (x 4) 0
x 4 0 OR x 4 0
Double Root
x 4 OR x 4
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METHOD 1 - FACTORING

• Trinomials Look for Reverse of Foil
• 4. 2x3 15x 7x2

2x3 7x2 15x 0
-3/2, 0, 5
(x) (2x2 7x 15) 0
(x) (2x 3)(x 5) 0
x 0 OR 2x 3 0 OR x 5 0
x 0 OR x 3/2 OR x 5
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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• END OF METHOD 1
• FACTORING
• Click here to return to menu slide

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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.)

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METHOD 2 SQUARE ROOTS OF BOTH SIDES

1. x2 9

-3, 3
x ? 3
Note ? means both 3 and -3!
x -3 OR x 3
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METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 2. x2 18

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METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 3. x2 9

Cannot take a square root of a negative. There
are NO real number solutions!
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METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 4. (x-2)2 9

-1, 5
This means x 2 3 and x 2 3
x 5 and x 1
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METHOD 2 SQUARE ROOTS OF BOTH SIDES

• Rewrite as (glob)2 c first if necessary.
• 5. x2 10x 25 9

(x 5)2 9
2, 8
x 8 and x 2
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METHOD 2 SQUARE ROOTS OF BOTH SIDES

• Rewrite as (glob)2 c first if necessary.
• 6. x2 10x 25 48

(x 5)2 48
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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

1. x2 121

-11, 11
x ? 11
Note ? means both 11 and -11!
x -11 OR x 11
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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 2. x2 81

 
Square root of a negative, so there are NO real
number solutions!
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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

• Rewrite as (glob)2 c first if necessary.
• 3. 6x2 156

x2 26
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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 4. (a 7)2 3

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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

• Rewrite as (glob)2 c first if necessary.
• 5. 9(x2 14x 49) 4

(x 7)2 4/9
6?, 7?
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PRACTICE METHOD 2 SQUARE ROOTS OF BOTH SIDES

• 6.

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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• END OF METHOD 2
• SQUARE ROOTS OF BOTH SIDES
• Click here to return to menu slide

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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!

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METHOD 3 COMPLETE THE SQUARE

• Example 3x2 6 x2 12x

Simplify and write in standard form ax2 bx
c 0
2x2 12x 6 0
x2 6x 3 0
Set a 1 by division Note in some problems
a will already be equal to 1.
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METHOD 3 COMPLETE THE SQUARE
x2 6x 3 0
Move 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!
(x 3)2 12
Rewrite as (glob)2 c
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METHOD 3 COMPLETE THESQUARE
(x 3)2 12
Take square roots of both sides dont forget ?
Simplify
Solve for x
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PRACTICE METHOD 3 COMPLETE THE SQUARE

• Example 2b2 16b 6

Simplify and write in standard form ax2 bx
c 0
2b2 16b 6 0
b2 8b 3 0
Set a 1 by division Note in some problems
a will already be equal to 1.
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PRACTICE METHOD 3 COMPLETE THE SQUARE
b2 8b 3 0
Move 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!
(b 4)2 19
Rewrite as (glob)2 c
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PRACTICE METHOD 3 COMPLETE THE SQUARE
(b 4)2 19
Take square roots of both sides dont forget ?
Simplify
Solve for the variable
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PRACTICE METHOD 3 COMPLETE THE SQUARE

• Example 3n2 19n 1 n - 2

Simplify and write in standard form ax2 bx
c 0
3n2 18n 3 0
n2 6n 1 0
Set a 1 by division Note in some problems
a will already be equal to 1.
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PRACTICE METHOD 3 COMPLETE THE SQUARE
n2 6n 1 0
Move 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!
(n 3)2 8
Rewrite as (glob)2 c
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PRACTICE METHOD 3 COMPLETE THE SQUARE
(n 3)2 8
Take square roots of both sides dont forget ?
Simplify
Solve for the variable
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PRACTICE METHOD 3 COMPLETE THE SQUARE
What number completes each square?
1. x2 10x -3
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
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PRACTICE METHOD 3 COMPLETE THE SQUARE
Now rewrite as (glob)2 c
1. (x 5)2 22
1. x2 10x 25 -3 25
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
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PRACTICE METHOD 3 COMPLETE THE SQUARE
Show all steps to solve.
?k2 4k - ?
k2 12k - 2
k2 - 12k - 2
k2 - 12k 36 - 2 36
(k - 6)2 34
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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• END OF METHOD 3
• COMPLETE THE SQUARE
• Click here to return to menu slide

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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

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METHOD 4 QUADRATICFORMULA

• Use to solve 3x2 6 x2 12x

Standard Form 2x2 12x 6 0
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METHOD 4 QUADRATICFORMULA
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PRACTICE METHOD 4 QUADRATIC FORMULA
Show all steps to solve simplify.
2x2 x 6
2x2 x 6 0
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PRACTICE METHOD 4 QUADRATIC FORMULA
Show all steps to solve simplify.
x2 x 5 0
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PRACTICE METHOD 4 QUADRATIC FORMULA
Show all steps to solve simplify.
x2 2x - 4 0
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THE DISCRIMINANT MAKING PREDICTIONS
b2 4ac is called the discriminant
Four cases
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
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THE DISCRIMINANT MAKING PREDICTIONS
Use the discriminant to predict how many roots
each equation will have.
1. x2 7x 2 0
494(1)(-2)57 ?2 irrational roots
2. 0 2x2 3x 1
94(2)(1)1 ? 2 rational roots
3. 0 5x2 2x 3
44(5)(3)-56 ? no real roots
1004(1)(25)0 ? 1 rational double root
4. x2 10x 250
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THE DISCRIMINANT MAKING PREDICTIONS about
Parabolas
The zeros of a function are the x-intercepts on
its 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
14(2)(-6)49 ? 2 rational zeros opens up/vertex
below x-axis/2 x-intercepts
2. f(x) 2x2 x 6
14(2)(6)-47 ? no real zeros opens up/vertex
above x-axis/No x-intercepts
3. y -2x2 9x 6
814(-2)(6)129 ?2 irrational zeros opens
down/vertex above x-axis/2 x-intercepts
4. f(x) x2 6x 9
364(1)(9)0 ? one rational zero opens up/vertex
ON the x-axis/1 x-intercept
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THE DISCRIMINANT MAKING PREDICTIONS
Note the proper terminology The zeros of a
function are the x-intercepts on its 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 its graph.
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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• END OF METHOD 4
• QUADRATIC FORMULA
• Click here to return to menu slide

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FOUR METHODS HOW DO I CHOOSE?
Some suggestions Quadratic Formula works for
all quadratic equations, but look first for a
quicker method. Dont forget to simplify
square roots and use value of discriminant to
predict number of roots. Square Roots of Both
Sides use when the problem is easily written as
glob2 constant. Examples 3(x 2)212 or
x2 75 0
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FOUR METHODS HOW DO I CHOOSE?
Some suggestions Factoring doesnt always
work, but IF you see the factors, this is
probably the quickest method. Examples x2 8x
0 has a GCF 4x2 12x 9 0 is a PST x2
x 6 0 is easy to FOIL Complete the
Square best used when a 1 and b is even (so
you wont need to use fractions). Examples x2
6x 1 0
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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

• END OF HOW TO CHOOSE A
• QUADRATIC METHOD
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REVIEW QUADRATICFUNCTIONS

• The graph is a parabola. Opens up if a gt 0 and
  down if a lt 0.
• To find x-intercepts may have Zero, One or Two
  x-intercepts
• 1. Set y or "f(x)" to zero on one side of the
  equation
• 2. Factor use the Zero Product Prop to find
  TWO x-intercepts
• 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

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REVIEW QUADRATICFUNCTIONS
Example Problem f(x) x2 2x 8 a. Opens
UP since a 1 (that is, positive) b.
x-intercepts 0 x2 2x 8 0 (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
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PREVIEW/SUMMARY OF QUADRATIC EQUATIONS
FUNCTIONS

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