quadratic equations
DESCRIPTION
Using quadratic formula and the discriminantTRANSCRIPT
![Page 1: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/1.jpg)
Quadratic Formula
-b + b2 - 4ac
2a
ax2 + bx + c = 0
Rewrite quadratic equation in standard form a substitute a,b, and c
into formula.
x =
![Page 2: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/2.jpg)
Solve: 3x2 - 5 = 2x
![Page 3: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/3.jpg)
Step 1: Rewrite in standard form.
Solve: 3x2 - 5 = 2x
![Page 4: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/4.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
![Page 5: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/5.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
Step 2: Plug into formula and simplify.
![Page 6: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/6.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
x = -(-2) + (-2)2 - 4(3)(-5)
2(3)
Step 2: Plug into formula and simplify.
x = -b + b2 - 4ac
2a
![Page 7: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/7.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
x = -(-2) + (-2)2 - 4(3)(-5)
2(3)
Step 2: Plug into formula and simplify.
x =2 + 64
6
x = -b + b2 - 4ac
2a
![Page 8: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/8.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
x = -(-2) + (-2)2 - 4(3)(-5)
2(3)
Step 2: Plug into formula and simplify.
x =2 + 64
6
x = 2 + 86
x = -b + b2 - 4ac
2a
![Page 9: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/9.jpg)
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
x = -(-2) + (-2)2 - 4(3)(-5)
2(3)
Step 2: Plug into formula and simplify.
x =2 + 64
6
x = 2 + 86
= 5/3, -1
x = -b + b2 - 4ac
2a
![Page 10: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/10.jpg)
Solve: 3x2 = 5x - 2
![Page 11: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/11.jpg)
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
![Page 12: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/12.jpg)
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
x = -(-5) + (-5)2 - 4(3)(2)
2(3)
![Page 13: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/13.jpg)
![Page 14: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/14.jpg)
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
x = -(-5) + (-5)2 - 4(3)(2)
2(3)
x =5 + 1
6 = 1, 2/3
![Page 15: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/15.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2
answers.
d = b2 - 4ac
![Page 16: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/16.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2
answers.
d = b2 - 4ac
Where do you recognized this formula from?
![Page 17: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/17.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2
answers.
d = b2 - 4ac
Where do you recognized this formula from?
It’s part of the quadratic formula.
![Page 18: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/18.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number
solutions.
d = b2 - 4ac x = -b + b2 - 4ac
2aWhere do you
recognized this formula from?
It’s part of the quadratic formula.
If d = 0, then there is exactly one solution
![Page 19: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/19.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number
solutions.
d = b2 - 4ac x = -b + b2 - 4ac
2aWhere do you
recognized this formula from?
It’s part of the quadratic formula.
If d = 0, then there is exactly one solutionIf d > 0, then there are two
solutions.
![Page 20: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/20.jpg)
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2 real number
solutions.
d = b2 - 4acx =
-b + b2 - 4ac
2aWhere do you recognized this formula from?
It’s part of the quadratic formula.
If d = 0, then there is exactly one solutionIf d > 0, then there are two
solutions.If d < 0, then there are no real number solutions.
![Page 21: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/21.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
![Page 22: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/22.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
![Page 23: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/23.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
![Page 24: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/24.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
![Page 25: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/25.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
One solution
![Page 26: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/26.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
One solution
Let’s look at the equationgraphically.
![Page 27: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/27.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
One solution
Let’s look at the equationgraphically.
![Page 28: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/28.jpg)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
One solution
Let’s look at the equationgraphically.
Intersects x-axis only once.
![Page 29: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/29.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
![Page 30: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/30.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
![Page 31: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/31.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
![Page 32: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/32.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
No Real Number Solutions
![Page 33: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/33.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
No Real Number Solutions
Graphically:
![Page 34: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/34.jpg)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
No Real Number Solutions
Graphically:
Does not cross x-axis
![Page 35: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/35.jpg)
How many solutions does 3x2 - 2x - 1 = 0 have?
![Page 36: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/36.jpg)
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
![Page 37: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/37.jpg)
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = 4 + 12
d = 16
![Page 38: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/38.jpg)
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = 4 + 12
d = 16Two
solutions
![Page 39: Quadratic Equations](https://reader033.vdocuments.mx/reader033/viewer/2022061204/547f3265b4af9f210b8b4658/html5/thumbnails/39.jpg)
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = 4 + 12
d = 16Two
solutions
Graphically:
Crosses x-axis twice