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