Solving

Equations

02 cbxaxWhat are we going to do if we have non-zero values for a, b and c but can't factor the left hand side?

0362 xx This will not factor so we will complete the square and apply the square root method.

First get the constant term on the other side by subtracting 3 from both sides.362 xx

___ 3___ 62 xx

We are now going to add a number to the left side so it will factor into a perfect square. This means that it will factor into two identical factors. If we add a number to one side of the equation, we need to add it to the other to keep the equation true.

Let's add 9. Right now we'll see that it works and then we'll look at how to find it.

9 9 69 62 xx

69 62 xx Now factor the left hand side.

633 xx

two identical factors

63 2 xThis can be written as:

Now we'll get rid of the square by square rooting both sides.

63 2 x Remember you need both the positive and negative root!

63 x Subtract 3 from both sides to get x alone.

63 xThese are the answers in exact form. We can put them in a calculator to get two approximate answers.

55.063 x 45.563 x

Okay---so this works to solve the equation but how did we know to add 9 to both sides?

___ 3___ 62 xx 9 9

Before we noticed that the middle term was equal to:

21

21

2

2

cb

cb

cab

cab

Okay---so this works to solve the equation but how did we know to add 9 to both sides?

___ 3___ 62 xx 9 9

Before we noticed that the middle term

So to complete the square, the number to add to both sides is…

the middle term's coefficient divided by 2 and squared

(as long as a=1)

cb

cbcb

2

2

2

2

Let's solve another one by completing the square.

0182 xx

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 15444 2 xxxSquare root both sides (remember )

154 2 x

154 x 154 xAdd 4 to both sides to get x alone

2

28

the middle term's coefficient divided by 2 and squared

Let's solve another one by completing the square.

07102 xx

Since it doesn't factor get the constant on the other side ready to complete the square.

___7___102 xx

So what do we add to both sides?25

25 25

Factor the left hand side 32555 2 xxxSquare root both sides (remember )

325 2 x

245 x 245xSubtract 5 from both sides to get x alone

2

210

the middle term's coefficient divided by 2 and squared

You try:

• x2 - 8x + 3 = 0

• x2 -10x +7 = 0

• x2 +14x - 43 = 0