Factoring quadratic equation into binomials

Factoring quadratic equation into binomials:
    Remember how we factorized polynomials of the second degree? How do we factorize `x^2 + 7x + 12`?
12 = 1 x 12             1 + 12 = 13
12 = 2 x 6                2 + 6 =8

12 = 3 x 4      3 + 4 = 7

`x^2 + 7x +12` = `x^2 +(3 + 4)x + (3 x 4)`
                               = `(x+3)(x+4)`
There is another way of doing such factoring quadratic binomials. Supppose we want to find the factors of p(x) = `x^2 + 3x -10`. We know that if x - a factor of p(x), then p(a) = 0. In other words, a is a solution of the equation p(x) = 0;m that is, the equation `x^2+3x -10 =0` .
We know how to solve the quadratic equation `x^2 + 3x -10 = 0`. Using the quadratic formula

Factoring Quadratic Binomial Formula:

   The quadratic equation `ax^2 + bx + c = 0` of  factoring  formulae:

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

In the above problems,

                 x = -3 `+-` ` sqrt(3^2 - 4 .1 . (-10))/2`    =  (-3 `+-` `sqrt(59)` ) / 2 

                                                                             = (-3 `+-` 7)/ 2

                                                                             = 2 or -5

   Thus if we write p(x) = `x^2 + 3x -10 `, then p(2) = 0 and p(-5) = 0, so that by the Factor Theorem x-2 and x-(-5) = x + 5 are factors of p(x). By actual multiplication, we do find

                               (x - 2) (x +5) = `x^2 + 3x -10`

Examples for Factoring Quadratic Binomials:

Example 1:

Factoring the quadratic equation `6x^2-11x+3`into binomials.

Solution:

Given:

    `6x^2-11x+3`

To factor the above quadratic equation, we put the values for a, b, and c into the quadratic formula.

like this:

Calculating these expressions through, we get that

                                                  One root or solution is: 1.500
                                                  and the other root or solution is: 0.333
                                                  These answers have been rounded to 3 digits after the decimal point.

 Therefore, the required binomials are `(x - 1.500)` and `(x - 0.333)` 

Example 2:

Factoring the quadratic equation `x^2 -2x-1 =0` into binomials.

Solution:

To factor the above quadratic equation, we put the values for a, b, and c into the quadratic formula

like this:

Calculating these expressions through, we get that

One root or solution is: 2.414
and the other root or solution is: -0.414
These answers have been rounded to 3 digits after the decimal point.

Therefore the required binomials are `(x - 2.414)`  and `(x + 0.414)` .