How do you know if it's possible to factorise a quadratic expression?

Question

pottersheep · Answer

My teacher said something about "delta" and it being  a perfect square trinomial but maybe I heard wrong

pottersheep · Answer

also , is 4a^2 + 7ab - 2b factorable?

anonymous · Answer

http://www.wolframalpha.com/input/?i=4a^2+%2B+7ab+-+2b+

anonymous · Answer

Case 1.
$$x^2+bx+c$$is factorable if there exists factors of c that sum to b;

Example Case 1:
$$x^2−4x−12 $$is factorable because -6*2=-12 and -6+2=-4, thus$$x^2−4x−12=(x−6)(x+2)$$

anonymous · Answer

Case 2.
$$ax^2+bx+c $$is factorable if there exists factors of ac that sum to b;

Example Case 2:
3x2−5x−12
is factorable because ac=3(-12)=-36, and -9*4=-36 while -9+4=-5, thus$$3x^2−5x−12=3x^2-9x+4x-12=3x(x-3)+4(x-3)=(x-3)(3x+4)$$

anonymous · Answer

A method that spans all cases; the expression$$ax^2+bx+c$$(where a, b, and c are rational) is factorable as a perfect square trinomial if$$b^2-4ac=0$$or if$$b^2-4ac$$is a perfect square

anonymous · Answer

Case: Perfect Square Trinomial Example

$$4x^2+20x+25$$is a perfect square trinomial since$$b^2-4ac=20^2-4(4)(25)=0$$so it factors to$$(2x+5)^2$$

A practical way to determine if you have a perfect square trinomial is to notice that the first and the third terms of the quadratic expression are perfect squares.  In the above case$$4x^2;\ 25$$are both perfect squares.  Then note that$$(a+b)^2=a^2+2ab+b^2$$where$$a^2=4x^2;\ b^2=25 \implies a=2x;\ b=5$$so if $$2ab=2(2x)(5)=20x$$equals the middle term 20x (which it does) then you have perfect square trinomial

anonymous · Answer

$$b^2-4ac$$is called the discriminant, and may be what your teacher is calling "delta."  it is the radicand from the quadratic formula$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$Note that if the discriminant is a positive number, but not a perfect square, or if it is negative, then the quadratic expression is not factorable over the integers.