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Transcript of 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real...
![Page 1: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/1.jpg)
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A quadratic equation is an equation equivalent to one of the form
Where a, b, and c are real numbers and a 0 02 cbxax
To solve a quadratic equation we get it in the form above and see if it will factor.
652 xx Get form above by subtracting 5x and adding 6 to both sides to get zero on right side.-5x + 6 -5x + 6
0652 xx Factor of 6 {1,6 or 2,3}
023 xx Use the Null Factor law and set each factor = 0 and solve.
02or 03 xx 3x 2x
So if we have an equation in x and the highest power is 2, it is quadratic.
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In this form we could have the case where b = 0.
02 cbxaxRemember standard form for a quadratic equation is:
02 cax002 cxax
When this is the case, we get the x2 alone and then square root both sides.
062 2 x Get x2 alone by adding 6 to both sides and then dividing both sides by 2
+ 6 + 6
62 2 x2 2
32 x
Now take the square root of both sides remembering that you must consider both the positive and negative root.
3x Let's check: 0632
2 0632
2
066 066
Now take the square root of both sides remembering that you must consider both the positive and negative root.
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02 cbxaxWhat if in standard form, c = 0?
002 bxaxWe could factor by pulling an x out of each term.
032 2 xx Factor out the common x
032 xx Use the Null Factor law and set each
factor = 0 and solve.
032or 0 xx
2
3or 0 xx If you put either of these values in for
x in the original equation you can see it makes a true statement.
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2 4 5x x
1 0x 5 0x
1 5 0x x
1x 5x
4 5x x
2 4 5 0x x
Solving Quadratic Equations by Factoring
5
Example
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23 7 6x x 3 0x 3 2 0x
3 3 2 0x x
3x 2
3x
3 7 6x x
23 7 6 0x x 3 2x
6 :Factors of2, 31, 6
3:Factors of1, 3
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Example
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29 24 16x x 29 24 16 0x x
3 4 0x 3 4 3 4 0x x
4
3x
3 4x
9 16and are perfect squares
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Example
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32 18 0x x 2x
2 0x
2x
3x 3 0x 3 0x
3x 0x
2 9x 0
3x 3x 0
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Example
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23 3 20 7 0x x x
3x
3 0x
7x 7 0x 3 1 0x
1
3x
3x 3 1x
3:Factors of 1, 3 7 :Factors of 1, 7
7x 0 3 1x
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Example
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An alternate method of solving a quadratic equation is using the Principle of Taking the Square Root of Each Side of an Equation If x2 = a, then
x = + a
Solving Quadratic Equations by square roots
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Ex: Solve by taking square roots 3x2 – 36 = 0
First, isolate x2: 3x2 – 36 = 03x2 = 36
x2 = 12
Now take the square root of both sides:
12x 2 12x
32x x 2 2 3
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Ex: Solve by taking square roots 4(z – 3)2 = 100
First, isolate the squared factor:
4(z – 3)2 = 100(z – 3)2 = 25
Now take the square root of both sides:
25)3z( 2
253z z – 3 = + 5
z = 3 + 5
z = 3 + 5 = 8 and z = 3 – 5 = – 2 12
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Ex: Solve by taking square roots 5(x + 5)2 – 75 = 0
First, isolate the squared factor:
5(x + 5)2 = 75(x + 5)2 = 15
Now take the square root of both sides:
2( x 5 ) 15 x 5 15
x 5 15 x 5 15 , x 5 15
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Write the equation in the form Add to each side of the equation Factor the perfect-square trinomial
Take the square root of both sides of the equationSolve for x
Steps to solving Quadratic Equations by Completing the Square
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Ex: Solve x2 + 6x + 4 = 0 by completing the square
First, rewrite the equation with the constant on one side of the equals and a lead coefficient of 1.
x2 + 6x = – 4Add to both sides:
b =
6
6 = 32 = 9
x2 + 6x + 9 = – 4 + 9
x2 + 6x + 9 = 5
(x + 3)2 = 5
Now take the square root of both sides
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2( 3) 5x
3 5x
3 5x
{ 3 5, 3 5}x
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solve by completing the square.
02162 2 xx To complete the square we want the coefficient of the x2 term to be 1.
Divide everything by 20182 xx2 2 2 2
Since it doesn't factor get the constant on the other side ready to complete the square.
___1___82 xx
So what do we add to both sides?16
16 16
Factor the left hand side 24 15x
Square root both sides (remember ) 154 2 x
154 x 154xAdd 4 to both sides to get x alone
2
2
8
the middle term's coefficient divided by 2 and squared
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Ex: Solve 2y2 = 3 – 5y by completing the square
First, rewrite the equation with the constant on one side of the equals and a lead coefficient of 1.
2y2 + 5y = 3
Add [½(b)]2 to both sides: b =
(5/2)
5/2 [½(5/2)]2 = (5/4)2 = 25/16
y2 + (5/2)y + 25/16 = (3/2) + 25/16
y2 + (5/2)y + 25/16 = 24/16 + 25/16
(y + 5/4)2 = 49/16
Now take the square root of both sides
y2 + (5/2)y = (3/2)
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2( 5 / 4) 49 /16y
5 / 4 (7 / 4)y
(5 / 4) (7 / 4)y
y= - (5/4) + (7/4) = 2/4 = ½
and y= - (5/4) - (7/4) = -12/4 = - 3
y= { ½ , - 3}
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Ex: Solve 3n– 5 = (n – 1)(n – 2)
Is this a quadratic equation?
3n – 5 = n2 – 2n – n + 2
3n – 5 = n2 – 3n + 2
n2 – 6n + 7 = 0
Collect all terms
n2 – 6n = – 7 [½(-6)]2 = (-3)2 = 9
n2 – 6n + 9 = – 7 + 9
(n – 3)2 = 2
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2( 3) 2n
3 2n 3 2n
{3 2,3 2}n
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Consider a quadratic equation of the form ax2 + bx + c = 0 for (a≠0).
Completing the square
2ax bx c 2
b cx x
a a
2 2
2
2 2
b b c bx x
a 4a a 4a
Solving Quadratic Equations by Quadratic Formula
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Solutions to ax2 + bx + c = 0 for (a≠0) are:
22
2
b b 4acx
2a 4a
2b b 4acx
2a
2 2
2
2 2 2
b b 4ac bx x
a 4a 4a 4a
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WHY USE THE QUADRATIC FORMULA?
The quadratic formula allows you to solve
any quadratic equation, even if you cannot
factor it.
An important piece of the quadratic formula
is what’s under the radical:
b2 – 4ac
This piece is called the discriminant.
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WHY IS THE DISCRIMINANT IMPORTANT?
The discriminant will take on a value that is positive,
0, or negative.
The value of the discriminant indicates two distinct
real solutions, one real solution, or no real solutions,
respectively. 25
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WHAT THE DISCRIMINANT TELLS YOU!
Value of the Discriminant Nature of the Solutions
<0 Negative 2 imaginary solutions
=0 Zero 2- Double Solution
>0 Positive – perfect square 2 Real's- Rational
>0 Positive – non-perfect square 2 Real's- Irrational
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Use the discriminant to determine the number and
type of solutions for the following equation.
5 – 4x + 12x2 = 0
a = 12, b = –4, and c = 5b2 – 4ac = (–4)2 – 4(12)(5)
= 16 – 240 = –224<0
There are no real solutions.
Example
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6 – 3x2 –2x = 0
a = – 3, b = –2, and c = 6b2 – 4ac = (–2)2 – 4(– 3)(6)
= 4 + 72 = 76>0
There are 2 distinct root .
Example
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29
x2 –4x +4= 0
a = 1, b = – 4, and c = 4b2 – 4ac = (–4)2 – 4(1)(4)
= 16 – 16 = 0
We have two double root
Example
![Page 30: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/30.jpg)
By completing the square on a general quadratic equation in standard form we come up with what is called the quadratic formula.
a
acbbx
2
42
This formula can be used to solve any quadratic equation whether it factors or not. If it factors, it is generally easier to factor---but this formula would give you the solutions as well.
We solved this by completing the square but let's solve it using the quadratic formula
a
acbbx
2
42
1
(1)
(1)
6 6 (3)2
12366
Don't make a mistake with order of operations! Let's do the power and the multiplying first.
02 cbxax
0362 xx
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2
12366 x
2
246
626424
2
626
2
632
There's a 2 in common in the terms of the numerator
63 These are the solutions we got when we completed the square on this problem.
NOTE: When using this formula if you've simplified under the radical and end up with a negative, there are no real solutions.(There are complex (imaginary) solutions, but that will be dealt with in year 12 Calculus).
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ax2 + bx + c = 0 for (a≠0).
If,then
If , then we have two complex number (no-Real –Root)
If ,then
2
2
b b 4ac bx'
2a 2a
b b 4ac bandx"
2a 2a
bx' x"
2a
=b2 – 4ac
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If,then
If ,then we have two complex number (no-Real –Root)
If ,then
b' b' ac b' 'x'
a a
b' b' ac b' 'andx"
a a
b'x' x"
a
=–ac where =
If b is an even number in the quadratic Equation ax2 + bx + c = 0 for
(a≠0).
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The sum of the roots of the Quadratic equation:
ax2 + bx + c = 0 is
And the product of roots is
This means that:
Sum and Product of the Root
𝑺=−𝒃𝒂
{𝑥′+𝑥 ′ ′=𝑆=−𝑏𝑎
𝑥 ′ .𝑥 ′ ′=𝑃=𝑐𝑎
P
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Note:Two numbers whose given sum is S and given product is P are roots of the equation x2 - Sx + P= 0 Example:
Find two numbers knowing their sum S and their product P :
S=4 and P=4
The two required numbers are the solutions of the equation: x2 - 4x + 4= 0 ,that is (x-2)2
=0,which gives x’=x”=2
![Page 36: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/36.jpg)
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Example
Write the form of a quadratic equation whose roots are x’= and x”=Solution:The sum S=x’+x”=
And the product P=x’.x”=
The equation :
![Page 37: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/37.jpg)
Example
22 7 11 0x x
2
2
4
2
7 7 4(2)( 11
2, 7, 11
7 137 2 Reals - Irr
)
2
ational4
(2)
b b ac
b
a
a c
Solve using the Quadratic Formula
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Ex: Use the Quadratic Formula to solve x2 + 7x + 6 = 0
Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by
a2ac4bb
x2
Identify a, b, and c in ax2 + bx + c = 0:
a = b = c = 1 7 6
Now evaluate the quadratic formula at the identified values of a, b, and c
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)1(2)6)(1(477
x2
224497
x
2257
x
257
x
x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6
x = { - 1, - 6 }
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Ex: Use the Quadratic Formula to solve
2m2 + m – 10 = 0
Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by
a
acbbm
2
42
Identify a, b, and c in am2 + bm + c = 0:
a = b = c = 2 1 - 10
Now evaluate the quadratic formula at the identified values of a, b, and c
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)2(2)10)(2(411
m2
48011
m
4811
m
491
m
m = ( - 1 + 9)/4 = 2 and m = (-1 – 9)/4 = - 5/2
m = { 2, - 5/2 }
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42
Solve 11n2 – 9n = 1 by the quadratic formula.11n2 – 9n – 1 = 0, soa = 11, b = -9, c = -1
)11(2
)1)(11(4)9(9 2
n
22
44819
22
1259
22
559
Example
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43
)1(2
)20)(1(4)8(8 2
x
2
80648
2
1448
2
128 20 4 or , 10 or 22 2
x2 + 8x – 20 = 0 (multiply both sides by 8)
a = 1, b = 8, c = 20
8
1
2
5Solve x2 + x – = 0 by the quadratic formula.
Example
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44
Solve x(x + 6) = 30 by the quadratic formula.x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
)1(2
)30)(1(4)6(6 2
x
2
120366
2
846
So there is no real solution.
Example
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45
Solve 12x = 4x2 + 4. 0 = 4x2 – 12x + 4 0 = 4(x2 – 3x + 1)
Let a = 1, b = -3, c = 1
)1(2
)1)(1(4)3(3 2
x
2
493
2
53
Example
![Page 46: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/46.jpg)
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Solve the following quadratic equation.
02
1
8
5 2 mm
0485 2 mm
0)2)(25( mm
02025 mm or
25
2 mm or
Example
![Page 47: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/47.jpg)
Solving Quadratic Equations by the Quadratic Formula
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1) { 9,7}
2) {6, 14}
3) { 3,8}
7 3 7 34) ,
2 2
5 47 5 475) ,
6 6
x
x
x
i ix
i ix
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![Page 48: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/48.jpg)
0
A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 216 64.h t How long does it take for the diver to hit the surface of the water?
0 0
2 0t 2 0t 2t 2t seconds
216 64t 16 2 4t
16 2t 2t
Quadratic Equations and Problem Solving
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![Page 49: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/49.jpg)
5 176w w
The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?
11w The width is w.
11 0w 11w
The length is w+5.l w A
2 5 176w w 2 5 176 0w w
16 0w 16w
11w 11 5l 16l
feet
feet
176 :Factors of1,176 2, 88 4, 44
8, 22 11,16
16w 0
Quadratic Equations and Problem Solving
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![Page 50: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/50.jpg)
x
Find two consecutive odd numbers whose product is 23 more than their sum?
Consecutive odd numbers: x
5x 5x 2 2 2 25x x x
2 25 0x 5x
5 0x 5 0x
5, 3 5, 7
5 2 3 5 2 7
2.x 2x 2x x 23
2 22 2 2 25xx x x x
2 25 2525x
2 25x
5x 0
Quadratic Equations and Problem Solving
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![Page 51: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/51.jpg)
a x
The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?
12a
.Pythagorean Th
22 27 13x x
5x
5
meters
7b x 13c
2 2 14 49 169x x x 22 14 120 0x x 22 7 60 0x x
2
5 0x 12 0x
12x
12 7b meters
2 2 2a b c
60 :Factors of 1, 60 2, 303, 20 4,15 5,12
5x 12x 0
6,10
Quadratic Equations and Problem Solving
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![Page 52: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/52.jpg)
2x
The square of a number minus twice the number is 63. Find the number.
7x
7x
x is the number.
2 2 63 0x x
7 0x 9 0x
9x
2x 63
63:Factors of 1, 63 3, 21 7, 9
9x 0
Quadratic Equations and Problem Solving
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![Page 53: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/53.jpg)
SUMMARY OF SOLVING QUADRATIC EQUATIONS
• Get the equation in standard form: 02 cbxax
• If there is no middle term (b = 0) then get the x2 alone and square root both sides (if you get a negative under the square root there are no real solutions).• If there is no constant term (c = 0) then factor out the common x and use the null factor law to solve (set each factor = 0).
• If a, b and c are non-zero, see if you can factor and use the null factor law to solve.
• If it doesn't factor or is hard to factor, use the quadratic formula to solve (if you get a negative under the square root there are no real solutions).
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![Page 54: 1. A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a 0 To solve a quadratic equation we get.](https://reader033.fdocuments.net/reader033/viewer/2022052913/5697bf991a28abf838c91621/html5/thumbnails/54.jpg)
a
acbbxcbxax
2
40
22
If we have a quadratic equation and are considering solutions from the real number system, using the quadratic formula, one of three things can happen.
3. The "stuff" under the square root can be negative and we'd get no real solutions.
The "stuff" under the square root is called the discriminant.
This "discriminates" or tells us what type of solutions we'll have.
1. The "stuff" under the square root can be positive and we'd get two unequal real solutions 04 if 2 acb
2. The "stuff" under the square root can be zero and we'd get one solution (called a repeated or double root because it would factor into two equal factors, each giving us the same solution).
2if 4 0b ac
04 if 2 acb
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