Question

What is the new Transforming Method to solve quadratic equations?

Answer 1

Say for instance you have...

`x^2+bx`

This can be transformed into:

`(x+b/2)^2-(b/2)^2`

Let's find out if the expression above translates back into `x^2+bx`...

`(x+b/2)^2-(b/2)^2`

`=({x+b/2}+b/2)({x+b/2}-b/2)`

`=(x+2*b/2)x`

`=x(x+b)`

`=x^2+bx`

The answer is YES.

Now, it is important to note that `x^2-bx` (notice the minus sign) can be transformed into:

`(x-b/2)^2-(b/2)^2`

What you are doing here is completing the square . You can solve many quadratic problems by completing the square.

Here is one primary example of this method at work:

`ax^2+bx+c=0`

`ax^2+bx=-c`

`1/a*(ax^2+bx)=1/a*-c`

`x^2+b/a*x=-c/a`

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

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

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

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

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

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

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

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

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

The famous quadratic formula can be derived by completing the square .

Answer 2

The new Transforming Method to solve quadratic equations.
CASE 1. Solving type `x^2 + bx + c = 0`. Solving means finding 2 numbers knowing their sum (`-b`) and their product (`c`). The new method composes factor pairs of (`c`), and in the same time, applies the Rule of Signs. Then, it finds the pair whose sum equals to (`b`) or (`-b`).
Example 1. Solve `x^2 - 11x - 102 = 0`.
Solution. Compose factor pairs of `c = -102`. Roots have different signs. Proceed: `(-1, 102)(-2, 51)(-3, 34)(-6, 17).` The last sum `(-6 + 17 = 11 = -b).` Then the 2 real roots are: `-6` and `17`. No factoring by grouping.
CASE 2 . Solving standard type: `ax^2 + bx + c = 0` (1).
The new method transforms this equation (1) to: `x^2 + bx + a*c = 0` (2).
Solve the equation (2) like we did in CASE 1 to get the 2 real roots `y_1` and `y_2`. Next, divide `y_1` and `y_2` by the coefficient a to get the 2 real roots `x_1` and `x_2` of original equation (1).
Example 2. Solve `15x^2 - 53x + 16 = 0`. (1) `[a*c = 15(16) = 240].`
Transformed equation: `x^2 - 53 + 240 = 0` (2). Solve equation (2). Both roots are positive (Rule of Signs). Compose factor pairs of `a*c = 240`. Proceed: `(1, 240)(2, 120)(3, 80)(4, 60)(5, 48)`. This last sum is `(5 + 48 = 53 = -b)`. Then, the 2 real roots are: `y_1 = 5` and
`y_2 = 48`. Back to original equation (1), the 2 real roots are: `x_1 = y_1/a = 5/15 = 1/3;` and `x_2 = y_2/a = 48/15 = 16/5.` No factoring and solving binomials.

The advantages of the new Transforming Method are: simple, fast, systematic, no guessing, no factoring by grouping and no solving binomials.