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Solving (Monic) Quadratic Equations

The quick and dirty methods you may have learned simplified!

2 0

1

bx

a

a cx

In solving quadratic equations, there are three ways in which to solve for x:

- by square roots (directly)- by factoring- by completing the square- by the quadratic formula

**Note: in this presentation, we assume all quadratic equations are monic, meaning . If this is not the case, divide all terms by “a” to make it monic.

2 0bx cax

1a

-by square roots-by factoring-by completing the square-by the quadratic formula

One of the easiest cases is if in the quadratic equation, there is no “bx” or (linear term) appearing!

If this is the case, all that needs to be done is to

- Isolate the x term on one side of the equation

- Square root both sides!- Remember the plus and minus!!

-by square roots-by factoring-by completing the square-by the quadratic formula

2 16 0x

No “bx” appearing!

-by square roots-by factoring-by completing the square-by the quadratic formula

2 16 16 0 16x

Isolate the x term….

2 16 0x

2 16x

-by square roots-by factoring-by completing the square-by the quadratic formula

2 16x

Square root both sides….

2 16x

-by square roots-by factoring-by completing the square-by the quadratic formula

2 16x Remember the plus and minus….

4x

-by square roots-by factoring-by completing the square-by the quadratic formula

The next method is to factor! This can be tricky and sometimes near impossible, but if it works, it can save you a lot of time!!

To factor,- Look at your “c” term and determine what

numbers multiply together to become that number

- With the numbers from the first step, determine if any combination of the numbers when added together result in “b”.

- These numbers from the second step (if it works) now need to be written as (x-(first number) )x(x-(second number))=0

- To solve this, set each part, i.e. one part is (x-(first number) ), equal to zero individually and solve for x.

-by square roots-by factoring-by completing the square-by the quadratic formula

2 6 0x x

This will factor nicely!

-by square roots-by factoring-by completing the square-by the quadratic formula

2 6 0x x Determine the numbers that multiply together to be equal to -6:

·3 2· 3 6· 1 66 12 ·

-by square roots-by factoring-by completing the square-by the quadratic formula

2 6 0x x

·3 2· 3 6· 1 66 12 ·

Now look at the sum of each of these pairs of numbers. If one pair is equal to -1=b, those are the numbers used to factor the expression

1 1 5 5

-by square roots-by factoring-by completing the square-by the quadratic formula

2 6 0x x

( (2))·( ( 3)) 0

( 2)·( 3) 0

x x

x x

Having identified our factors, we can write the above expression as….

-by square roots-by factoring-by completing the square-by the quadratic formula

( 2)·( 3) 0x x

2 0

2 2 0 2

2

x

x

x

Lastly, we set each part (factor) equal to zero and solve for x:

3 0

3 3 0 3

3

x

x

x

-by square roots-by factoring-by completing the square-by the quadratic formula

A third method is called completing the square. In this method, - In the given form, make all the terms containing x

appear on one side of the equation and all the “other” terms (those not containing x) appear on the other side

- Identify the “bx” term in your equation and add to both sides of the equation.

- After this point, the side containing the x terms will factor as

- To finish the problem, square root both sides and

subsequently solve for x.

2

2

b

2

2xb

-by square roots-by factoring-by completing the square-by the quadratic formula

2 12 28 0xx We can try and factor this (and if we are careful, it will work) but we are going to try using the completing the square method

-by square roots-by factoring-by completing the square-by the quadratic formula

2 12 28 0xx First we move the x terms to one side of the equation and the other terms to the other side of the equation….

2

2

12 28 28 0 28

12 28

x

x

x

x

-by square roots-by factoring-by completing the square-by the quadratic formula

2 12 28x x Next, we determine our b value and compute And add it to each side of the equation.

2

2

b

2 2212

12 ( 6) 32 2

6b

b

So,

2

2

12 36 28 36

12 36 64

x

x

x

x

-by square roots-by factoring-by completing the square-by the quadratic formula

2 12 36 64xx After this, the left side of the equation will factor (this is left up to you to verify) as

2

2

12 36 64

6 64

x x

x

2

2xb

-by square roots-by factoring-by completing the square-by the quadratic formula

26 64x

Lastly, finish solving for x using square roots and other basic algebra skills:

26 64

6 8

6 6 8 6

8 6 or 8 6

14 or 2

x

x

x

x x

x x

-by square roots-by factoring-by completing the square-by the quadratic formula

A fourth method, and one that will always work is the quadratic formula. This method comes out of completing the square:- Determine a, b, and c in your quadratic

equation. (Note: in this presentation we are assuming that a=1.)

- Substitute in the appropriate values into the following formula and simplify to obtain your solutions:

(This is the quadratic formula)

2 4

2x

b b ac

a

-by square roots-by factoring-by completing the square-by the quadratic formula

2 4

2x

b b ac

a

2 7 9 0xx This problem can be solved by completing the square, but the quadratic formula is going to be used to determine the x values.

-by square roots-by factoring-by completing the square-by the quadratic formula

2 4

2x

b b ac

a

2 7 9 0xx First we determine our a, b, and c:

1 7 9a b c

-by square roots-by factoring-by completing the square-by the quadratic formula

22 ( 7) ( 7) 4(1)(9)4

2 2(1)

17 49 3

2 2

36 7

b c

ax

b a

Then we use the quadratic formula with these values:

1 7 9a b c

Thus

7 7 or

13 13

2 2x x