quadratic equations
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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 =
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.
Solve: 3x2 - 5 = 2x
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
Solve: 3x2 - 5 = 2x
Step 1: Rewrite in standard form.3x2 - 2x - 5 = 0
a b c
Step 2: Plug into formula and simplify.
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
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
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
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
Solve: 3x2 = 5x - 2
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
x = -(-5) + (-5)2 - 4(3)(2)
2(3)
Solve: 3x2 = 5x - 2
3x2 - 5x + 2 = 0
x = -(-5) + (-5)2 - 4(3)(2)
2(3)
x =5 + 1
6 = 1, 2/3
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2
answers.
d = b2 - 4ac
Discriminant - used to determine whether the quadratic equation has 0, 1, or 2
answers.
d = b2 - 4ac
Where do you recognized this formula from?
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.
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
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.
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.
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
How many solutions does x2 - 2x + 1 = 0 have?
d = b2 - 4ac
d = (-2)2 - 4(1)(1)
d = 4 - 4
d = 0
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
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.
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.
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.
How many solutions does 2x2 - 2x + 1 = 0 have?
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
How many solutions does 2x2 - 2x + 1 = 0 have?
d = (-2)2 - 4(2)(1)
d = 4 - 8
d = -4
No Real Number Solutions
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:
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
How many solutions does 3x2 - 2x - 1 = 0 have?
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = 4 + 12
d = 16
d = (-2)2 - 4(3)(-1)
How many solutions does 3x2 - 2x - 1 = 0 have?
d = 4 + 12
d = 16Two
solutions
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
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