What is the quadratic formula?

Do Now:

Solve by completing the square

x2+8x+15=0

I know thequadratic formula,

do you?

Is there some way we can get the same result without all of the work?

Yes, but to get our general formula, we are going to have to complete the square one more time.

This time, we are going to complete the square using the standard form of the quadratic equation.

This will give us a general solution for x

Here we go…

Write in standard form

Get the x’s alone on one side

Divide by a.€

ax 2 + bx + c = 0

€

ax 2 + bx = −c

€

x 2 +b

ax =

−c

a

The hard part…

Take half of “b” and square it, add this number to both sides.

Factor the left, and add the right€

x 2 +b

ax +

b2

4a2=b2

4a2−c

a

€

x +b

2a

⎛

⎝ ⎜

⎞

⎠ ⎟2

=b2 − 4ac

4a2

Almost done…

Take the square root of both sides

Solve for x

€

x +b

2a

⎛

⎝ ⎜

⎞

⎠ ⎟2

= ±b2 − 4ac

4a2

€

x +b

2a

⎛

⎝ ⎜

⎞

⎠ ⎟= ±

b2 − 4ac

2a

€

x = −b

2a±b2 − 4ac

2a

Final form?

This is known as the QUADRATIC FORMULA.

Instead of going through the process of completing the square we can just use this formula.

€

x =−b ± b2 − 4ac

2a

How do we use the quadratic formula?

First we have to make sure our equation is written in STANDARD FORM. If the order is wrong you could get confused.

Plug the coefficients in the correct locations

Example

2x2-3x-2=0 a=2, b=-3, c=-2

€

x =−b ± b2 − 4ac

2a

€

x =−(−3) ± (−3)2 − 4(2)(−2)

2(2)

€

=3± 9 +16

4

€

=3± 25

4=

3± 5

4

€

x =3 + 5

4,

3− 5

4⇒

8

4,

−2

4

€

x = 2, −1

2

One more for me

€

x − 6 = −11

x

Try on your own…

€

4x 2 −12x + 25 = 0

Review

We got the quadratic formula by completing the square

We need to remember to have the equation in standard form

Summary/HW

Of the methods we have learned, which do you prefer and why?

HW pg 81, 1-15 odd