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When solving a quadratic equation by factoring, what property do you need to use.

Demonstrate how would you solve a quadratic equation with factoring and provide a quadratic equation for your classmate to solve by factoring

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
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Given  :
aX2 + bx + c

to choose 2 numbers such as m. in such a way that

m+ n = - b/a
m. n = c/a

If this equation has solution with integer values, then equation can be factored to

a X2 + bX + c = a( x -m )( x-n)

Example: a =1:

X - 23X + 38 =

2 numbers ( m.n) whose sum is 23 and product 38 are ( 19 , 4), therefore:

X- 23X + 76 = (X -19) ( X-4)

In order for a quadratic to be factorable to binomial with integer values.

It is necessary and sufficient that -b to be Sum of the products of 2 factors of C:

In here we write the number 38 as product of its prime factors as:

76= 2 . 2 . 19   factors are (2, 4, 19)

we see that 76 = 4 *19    4+ 19 =23 , therefore the quadratic is factorable as above.

Now consider the following:

3 X2 - 14 X + 15 =

Now we have to have 2 numbers is 14, and product is 45.

The numbers by calculating in our mind are (5,9)

We break up the 14 into 5+9

3X2  - 9X -5X +15 =

We factor by part:

3X( X -3 ) - 5 ( X - 3) =
Now Common factor is ( X-3) , therefor

3 X- 14X + 15 = ( X- 3) ( 3X - 5 )

If :
X2  + 5X - 7
Here there is no 2 integers where their Sum is +5, and product is -7, in this case we
factor by what is called : Factoring by Completing square which gives us factors of
irrational and complex numbers.