Solving a quadratic equation by factorization

A quadratic equation is an equation of the form ax^2+bx+c=0, where a, b, and c are known constants and a  i not zero e.g. 2x^2+3x+7=0; 3x^2-4x=0; 7x^2-9=0 etc.

To solve by factorization, first factorize the LHS i.e. ax^2+bx+c, by first finding two factors which i will call T1 and T2 such that; (i) their sum is b i.e. T1+T2=b and (ii) their product is ac i.e. T1*T2=ac. Next replace bx with T1x and T2x to obtain ax^2+T1x+T2x+c=0. Then carry out group factorization of the LHS.

I will next demonstrate using two examples;
Example 1
Solve by factorization; x^2+7x+12=0
Solution
In this example our a=1, b=7 and c=12. Hence we need two factors T1 and T2 such that;
(i) T1+T2=7 and (ii) T1*T2=12. The two factors are T1=3 and T2= 4.
Next we replace 7x with 3x and 4x to get;
x^2+3x+4x+12=0 ......(***)
The next step is to factorize the LHS by group factorization i.e. we take the first two terms x^2 and 3x and factorize [i.e. x^2+3x=x(x+3)] and then factorize the next two terms i.e. [4x+12=4(x+3)]
Such that our equation (***) above becomes;
x^2+3x+4x+12=x(x+3)+4(x+3)=0
Note that (x+3) is common thus we have;
(x+3)(x+4)=0
(x+3) and (x+4) are the two factors of x^2+7x+12. Since their product is 0 it implies that either;
x+3=0 or x+4=0
Hence x=-3 or x=-4.
-3 and -4 are the solutions or the roots of the quadratic equation x^2+7x+12=0
Example 2.
solve 3x^2+10x+8=0
Solution
In this example our a=3, b=10 and c=8
Hence we need two factors T1 and T2 such that (i) T1+T2=10 and (ii) T1*T2=24
The two factors are 4 and 6.

Next we replace 10x with 6x and 4x to get;

3x^2+6x+4x+8=0 ......(******)

The next step is to factorize the LHS by group factorization i.e. we take the first two terms 3x^2 and 6x and factorize [i.e.3x^2+6x=3x(x+2)] and then factorize the next two terms i.e. [4x+8=4(x+2)] 

Such that our equation (******) above becomes;

3x^2+6x+4x+8=3x(x+2)+4(x+2)=0

Note that (x+2) is common thus we have;

(x+2)(3x+4)=0

(x+2) and (3x+4) are the two factors of 3x^2+8x+10. Since their product is 0 it implies that either;

x+2=0 or 3x+4=0

Hence x=-2 or x=-4/3. 

-2 and -4/3 are the solutions or the roots of the quadratic equation 3x^2+8x+10=0