# Explain how to factor x^2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation.  In at least 150 words.

tiburtius | Certified Educator

Any polynomial `p(x) = x^n+a_1x^(n-1)+cdots+a_(n-1)x+a_n`  of degree `n` can be factored like this

`p(x)=(x-x_1)^(m_1)(x-x_2)^(m_2)cdots(x-x_r)^(m_r)`

where `x_1,x_2,ldots,x_r` are roots of equation `p(x)=0` and numbers `m_1,m_2,ldots,m_r` are multiplicities of each of the roots.

Hence in order to factor your expression you need to find roots of equation

`x^2+bx+c=0`                                                          (1)

Since this is quadratic equation its roots are

`x_(1,2)=(-b pm sqrt(b^2-4ac))/(2a)`

`x_(1,2)=(-b pm sqrt(b^2-4c))/2`

Also notice that if `b^2-4c=0` we will have only one root but its multiplicity will be 2. In that case your solution would be

`x^2+bx+c=(x+b/2)^2`

In general case your solution would be

`x^2+bx+c=(x-(-b - sqrt(b^2-4c))/2)(x-(-b + sqrt(b^2-4c))/2)`

You colud also use Viete's formulas to find solutions to equation (1) or you could simply guess if  you have nice numbers, but the way showed above works always and it's not that hard once you use real numbers instead of general ones.

oldnick | Student

`ax^2+bx+c= a(x^2+b/ax +c/a)`

Now we want write `x^2 +b/a x +c/a`  as a square  adding d i.e:

`(x+y)^2=x^2+b/a x +c/a+d`

`x^2+2xy +y^2= x^2+b/ax+c/a +d`

so: `2xy=b/a x`     `rArr` `y=b/2a`   and `c/a+d=b^2/(4a^2)`  `d=b^2/(4a^2)-c/a=`

Then:    `d=(b^2-4ac)/(4a^2)`

we have got:    `a(x^2+b/a x+ c/a)=a[(x+b/2a)^2-(b^2-4ac)/(4a^2)]`

From   `(A^2-B^2)=(A+B)(A-B) ` :

`a(x^2+b/a x+c/a)=a(x+b/(2a) +1/(2a) sqrt(b^2-4ac))(x+b/(2a) -1/(2a) sqrt(b^2-4ac))=`

`=1/(4a) (2ax+b +sqrt(b^2-4ac))(2ax+b-sqrt(b^2-4ac))`

THIS  FACTORIZIGIN IS UNIQUE.