How to factor out polynomial into binomials

Question

Ex:

 

-24x2+48x+74=2

-24x2+48x+72=0

?

 

Ex:

3x2-30x+65=2

3x2-30x+63=0

?

Answer 1

Hi Cor;

-24x²+48x+72=0

Both coefficients 24 and 48, as well as the constant 72, are multiples of 24.  Henceforth, 24 can be factored out...

-24(x²-2x-3)=0

Let's divide both sides by -24...

[-24(x²-2x-3)]=0/-24

x²-2x-3=0

I did this to illustrate my point that the coefficient of -24 is no longer necessary to resolve this.

Let's factor.

For the FOIL...

FIRST must be (x)(x)=x²

OUTER and INNER must add-up to-2x.

(x-1)(x-1)

Let's FOIL...

FIRST...(x)(x)=x²

OUTER...(x)(-1)=-1x=-x

INNER...(-1)(x)=-1x=-x

LAST...(-1)(-1)=1

x²-x-x+1

x²-2x+1

Obviously, we wanted the constant to be -3, not +1.  The difference is -4...

[(x-1)(x-1)]-4=0

(x-1)²=4

Let's square root both sides..

√[(x-1)²=√4

x-1=2

x=3

Because (x-1)² has the same result as [-(x-1)²], we must consider such...

-(x-1)=2

-x+1=2

x-1=-2

x=-1

 

********************************

3x²-30x+63=0

3(x²-10x+21)=0

x²-10x+21=0

(x-7)(x-3)=0

x-7=0

x=7

x-3=0

x=3