factorize the following quadratic expession a^2-9

Question

i know how to factorize expressions. i know that to solve it you have to write a-9 twice because its to the second power, however im confused as to how to get two numbers to multiply that will give me -9 and add them together to get -9 as well....

Answer 1

The process you are trying to use is called the sum and product method. Briefly, if we have a quadratic expression of the form ax² + bx + c then according to the sum and product method, you have to find two numbers whose sum is equal to b and product is equal to c. Those two numbers will be the roots to the quadratic expression, given that a is equal to 1. It should be clear to you in the following example:

Quadratic Expression: x²+5x + 6

1. Identify a, b and c. (a is the coefficient of x2, b is the coefficient of x and c is the constant number that follows)
• a = 1
• b = 5
• c = 6.

2. Two numbers whose sum = 5 will be:
• 0+5 OR
• 1+4 OR
• 2+3

3. Try to multiply each of those two numbers to get c (which is equal to 6) you will find out that only 2 & 3 work. Their sum is 5 = b and there product is 6 = c

4. Now you will get your expression factorized into: (x + 2)(x +3). If you try to expand this expression you will get the initial quadratic expression of x² + 5x + 6

Now after that introduction I will explain your case. You have x2 – 9. Let us follow the steps we mentioned above

1. Identify a, b and c.
• a = 1
• b = 0 (Remember b is the number multiplied by x. You have no x here which means that x is multiplied by 0)
• c = -9

2. Two numbers whose sum is equal to b:
• 0 and 0
• -1 and 1
• -2 and 2
• And the list goes on and on of all numbers with their additive inverses( or in simpler terms the same number with a negative sign)

3. Now you have to multiply the numbers you got to see which ones will give you a product of -9
• 0x0 = 0
• -2x2 = -4
• -3x3 = -9 BAM you got your product!
4. Now you will get your expression factorized into (x+3)(x-3). If you try to expand it you will get
x² – 9

 

What you did wrong is that you assumed since you have only two terms in your quadratic expression then both the sum and product should be equal to c. Always write your quadratic expression in the original ax² + bx + c and you will get the right answer so a² - 9 can be rewritten as: a² + 0a  - 9.

 

One last note: You can always apply the sum and product method to any quadratic expression. It will help you solve faster if you remember the algebraic identity which states that x² - m² can always factor into (x+m)(x-m).

It will help you further if you can memorize the first several perfect squares like:

4 = 2x2

9 = 3x3

16 = 4x4

25 = 5x5

36 = 6x6....

 

Wish that helps :)

 

Please let me know if you have any more questions :)

 

Robert Assaf

Answer 2

Jasmine:

 

Try to multiply (a + 3) and (a - 3).  Success guaranteed!

 

BTW, not a - 9 as you say.

 

Dr. G.

Answer 3

a^2-9

=(a^2)-(3^2)

=(a-3)(a+3)