What is the new Transforming Method to solve quadratic equations?

2 Answers
Apr 17, 2015

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 .

Apr 18, 2015

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.