Let Cbe a set containing all complex numbers and consider the two-variable polynomial f(x,y)=ax2+bxy+cy2 where {a,b,c}⊆C.
Since polynomials can be factored easier with an even number of terms and the polynomial f(x,y)has an odd number of terms, two numbers p and q can be defined where {p,q}⊆C and satistfying both conditions b=p+q⋅⋅⋅(1) and pa=cq...(2) to even the number of terms in f(x,y)whilst requiring a common factor between the first and last two terms.
(1):b=p+q⇒q=b−p...(1′)
f(x,y)=ax2+bxy+cy2=ax2+pxy+qxy+cy2=x(ax+py)+cy(cqx+y)=x(ax+py)+cy(pax+y)...(2)=x(ax+py)+pcy(ax+py)=(x+pcy)(ax+py)
The factorization procedure is not fully completed as the artifically defined constant p remains in the above result. Expressing p in non-artifically defined constants is neccessary as substituting the expression to the above result is a viable method.
(2):pa=cq⇒pq=ac...(2′)
(2′):p(b−p)=ac⇒−p2+bp−ac=0⇒p=−2−b±√b2−4ac
Sufficent formulas are solved and therefore, can be combined into a formula. However, the quadratic equation expressing p in non-artifically defined constants has two roots, which can be expressed using plus-minus signs as below. Please be reminded that the plus-minus signs must be either both plus or both minus for the formula to return a correct result, however the root with both the plus-minus signs are most likely to be considered because of avoidance of confusion in negative signs.
f(x,y)=(x+b±√b2−4ac2cy)(ax+2by±√b2−4acy)
When factorizing polynomials with only 1 variable, y=1 can be substitued into the formula, generating a new formula as below. Similiar to the baove formula, please be reminded that the plus-minus signs must be either both plus or both minus for the formula to return a correct result, however the root with both the plus-minus signs are most likely to be considered because of avoidance of confusion in negative signs.
f(x,y)=(x+b±√b2−4ac2c)(ax+2b±√b2−4ac)
Question 1: Factorize 2x2+5x+3.
2x2+5x+3=⎝⎛x+(5)+√(5)2−4(2)(3)2(3)⎠⎞⎝⎛(2)x+2(5)+√(5)2−4(2)(3)⎠⎞=(x+66)(2x+26)=(x+1)(2x+3)
Question 2: Factorize 42y2−6x2+36xy.
42y2−6x2+36xy=−6x2+36xy+42y2=⎝⎛x+(36)+√(36)2−4(−6)(42)2(42)y⎠⎞⎝⎛(−6)x+2(36)y+√(36)2−4(−6)(42)y⎠⎞\\=(x+47y)(−6x+24y)\\=−(x+7y)(x−y)
Question 3: Factorize x2+4x+2.
x2+4x+2=⎝⎛x+(4)+√(4)2−4(1)(2)2(2)⎠⎞⎝⎛(1)x+2(4)+√(4)2−4(1)(2)⎠⎞=(x+4+2√24)(x+24+2√2)=(x+2−√2)(x+2+√2)
Question 4: Factorize x2+xy+y.
x2+xy+y=⎝⎛x+(1)+√(1)2−4(1)(1)2(1)y⎠⎞⎝⎛(1)x+2(1)y+√(1)2−4(1)(1)y⎠⎞=(x+1+√3i2y)(x+2y+√3iy)=(x+2y−2√3iy)(x+2y+2√3iy)
The quadratic factorization method, also named QFM, outperforms ordinary methods such as the cross method in both aspects of limitations and effeciency.
