Quadratic Equations Prepared by Doron Shahar. Warm-up: page 15 A quadratic equation is an equation...
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Transcript of Quadratic Equations Prepared by Doron Shahar. Warm-up: page 15 A quadratic equation is an equation...
Warm-up: page 15A quadratic equation is an equation that can be written in the form _____________ where a, b, and c are constants and a ≠ 0.
The zero product property says that if , then either ________ or ________.
4:for Solve
149:Factor
)3)(2(:FOIL
2
2
xx
xx
xx
02 cbxax
0A0AB
0B
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FOIL and Factoring
)3)(2( xx 1492 xx
2x x3 x2 6
62 xx
)3)(2()32(2 xx
)2)(7( xx
1427 and 927
FirstOutsideInsideLast
FOIL Factor
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1.4.1 Solve by Factoring01492 xx
0)2)(7( xx
2or 7 xx
Starting Equation
Factor
Solution
0)2(or 0)7( xxZero Product Property
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1.4.2 Solve by Factoring
0982 xx
0)9)(1( xx
9or 1 xx
Starting Equation
Factor
Solution
0)9(or 0)1( xxZero Product Property
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Solve by Factoring6)3)(2( xx
0)3)(4( xx
3or 4 xx
Starting Equation
Factor
Solution
0)3(or 0)4( xxZero Product Property
662 xxFOIL6 6
0122 xx
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Intro to Completing the square42 x
42 x
2x
Starting Equation
Take square root of both sides of the equation
Solution
4xPlace ± on the right side of the equation
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Intro to Completing the square6)1( 2x
61or 61 xx
Starting Equation
Insert ± on right side
Solution
6 1 x
6)1( 2 xTake square root
61xGroup like terms1 1
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Goal of Completing the squareThe goal of completing the squares is to get a quadratic equation into the following form:
khx 2)(
khxkhx or
Starting Equation
Insert ± on right side
Solution
kx h
khx 2)(Take square root
khx Group like terms
h h
6)1( 2 xeg.
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1.4.5 Completing the square
03)1( 22
1 xStarting Equation3 3
3)1( 22
1 x
6)1( 2x
Add 3 to both sides
Multiply both sides by 2
Desired form
The goal of completing the squares is to get a quadratic equation into the following form: khx 2)(
32)1(2 22
1 x
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Example: Completing the square
08)3( 22
1 xStarting Equation8 8
8)3( 22
1 x
16)3( 2x
Add 8 to both sides
Multiply both sides by 2
Desired form
82)3(2 22
1 x
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Completing the square16)3( 2x
1or 7 xx
Equation from previous slide
Insert ± on right side
Solution
4 3 x
16)3( 2 xTake square root
43xGroup like terms
3 3
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1.4.2 General method of Completing the square
0982 xx
982 xx
1691682 xx
25)4( 2 x
Starting Equation
Add 9 to both sides
Add (−8/2)2=16 to both sides
Factor left side
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1.4.4 General method of Completing the square
823 2 xx
3
8
3
22 xx
9
1
3
8
9
1
3
22 xx
9
2523
1)( x
Starting Equation
Divide both sides by 3
Add ((−2/3)/2)2=1/9to both sides
Factor left side
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1.4.3 General method of Completing the square
02262 xx
2262 xx
922962 xx
13)3( 2 x
Starting Equation
Subtract 22 from both sides
Add (−6/2)2=9to both sides
Factor left side
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No solutions in quadratic equation
Solving by factoring works only if the equation has a solution.
Completing the square always works, and can be used to determine whether a quadratic equation has a solution.
13)3( 2 x
13)3( 2 x
Try solving
Take square root
¡PROBLEMA! You CANNOT take the square root of a negative number. Therefore, the equation has no solution.
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General method of Completing the square
02 cbxaxcbxax 2
22
2
22
a
b
a
c
a
b
a
b xx
22
22
)(a
b
a
c
a
bx
Starting Equation
Subtract c from both sides
Add ((b/a)/2)2=(b/2a)2
to both sides
Factor left side
Divide both sides by aa
c
a
b xx 2
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Quadratic Formula
a
acbbx
2
42
22
22
)(a
b
a
c
a
bx If we solve for in the previous equation, ,we get an equation called the quadratic formula.
x
Quadratic Formula
Starting Equation 02 cbxaxThe quadratic formula gives us the solutions to every quadratic equation.
a
acbbx
2
42 Solution
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Using the quadratic formulaStarting Equation 0169 2 xx
)9(2
)1)(9(4)6()6( 2 xSolution
9a 6b 02 cbxax1c
a
acbbx
2
42 Quadratic Formula
Plug 9 in for a, 6 for b, and 1 for c in the quadratic formula.
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Simplify your solution
)9(2
)1)(9(4)6()6( 2 xSimplify
18
06 x
18
06 x
18
6x
3
1xSolution
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1.4.1 Using the quadratic formulaStarting Equation 01492 xx
)1(2
)14)(1(4)9()9( 2 xSolution
1a 9b 02 cbxax14c
a
acbbx
2
42 Quadratic Formula
Plug 1 in for a, 9 for b, and 14 for c in the quadratic formula.
1
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Simplify your solution
)1(2
)14)(1(4)9()9( 2 xSimplify
2
259 x
2
59 x
2
59or
2
59
xx
7or 2 xxSolution
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1.4.3 Using the quadratic formulaStarting Equation 022 6 2 xx
)1(2
)22)(1(4)6()6( 2 xSolution
1a 6b 02 cbxax22c
a
acbbx
2
42 Quadratic Formula
Plug 1 in for a, −6 for b, and 22 for c in the quadratic formula.
1 +( )
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Simplify your solution
)1(2
)22)(1(4)6()6( 2 xSimplify
2
526 x
You cannot take the square root of a negative number. Therefore, there is no solution.
No Solution
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DiscriminantEquation Discrimina
nt# of Solutions
0169 2 xx
01492 xx
022 62 xx
02 cbxax acb 42
0)1)(9(4)6( 2
025)14)(1(4)9( 2
052)22)(1(4)6( 2
04 if Solutions No
04 if Solutions 2
04 ifSolution 1
2
2
2
acb
acb
acb
Solution 1
Solutions 2
Solutions
No
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CalculatorPut the Quadratic Formula program on your
calculator. Instructions are in the back of the class
notes.ORYou can come in to office hours to have me
load the program onto your calculator.Warning! The calculator will not always give you exact answers.
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Simplifying expressions withbaab xxx 2 then ,0 If
18
25 25 5
29 29 23
20 54 54 52
2
206
2
526 2
52
2
6 53
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Quadratic equations with DecimalsIf a quadratic equation has decimals, it is easiest to simply use
the quadratic formula. If you want, you can multiply both sides of the equation by a power of 10 (i.e., 10, 100, 1000, etc) to get rid of the decimals. This can make it easier to simplify the answer if you are evaluating the quadratic formula without a calculator.
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Quadratic equations with FractionsIf a quadratic equation has fractions (and does not factor), it is
often easy to simply use the quadratic formula. If you want, you can multiply both sides of the equation by the least common denominator to get rid of the fractions. This can make it easier to simplify the answer.
If a quadratic equation has fractions (and factors), it is often easier to factor after having gotten rid of the fractions.
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Variables in the denominatorsIf an equation has variables in the denominator, it is NOT a
quadratic equation. Such equations, however, can lead to linear equations.
We treat such equations like those with fractions. That is, we multiply both sides of the equation by a common denominator to get rid of the variables in the dominators. Ideally, we should multiply by the least common denominator.
Example: If our problem has B +16 in the denominator of one term,
and B in the denominator of another term, we multiply both sides of the equation by B(B+16). After the multiplication, the terms will have no variables in the denominators.
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Variables in the denominators1
15
16
15
BB
1)16(15
16
15)16(
BB
BBBB
)16()16(1515 BBBB
Starting Equation
Multiply both sides by B(B+16)
)16(15
)16(16
15)16(
BB
BBB
BBB
The problem is now in a form you can solve.
Distribute the B(B+16)
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Systems of equationsThere are two methods for solving systems of equations: Substitution and Elimination. Both work by combining the equations into a single equation with one variable. And sometimes the resulting equation leads to a quadratic equation.
We will only review substitution, because elimination is not a common method when working with systems of equations that lead to quadratic equations.
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