Question 1130128
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<pre>
Consider an equation

5a^2 - 7ab - 6b^2 = 0.       (1)



Divide both sides by b. You will get the equation

{{{5(a/b)^2 - 7(a/b) - 6}}} = 0.



Introduce new variable  x = {{{a/b}}}.  Then the last equation takes the form

5x^2 - 7x - 6 = 0.    (2)



Find its roots using the quadratic formula

{{{x[1,2]}}} = {{{(7 +- sqrt(7^2 -4*5(-6)))/(2*5)}}} = {{{(7 +- sqrt(169))/10}}} = {{{(7 +- 13)/10}}}.


So, the roots are  {{{-6/10}}} = {{{-3/5}}}  and  2.



It means that the polynomial (2) can be factored in this way

5x^2 - 7x - 6  = 5*(x-2)*(x+3/5) = (x-2)*(5x+3).



Returning to variables "a" and "b", you get

5a^2 - 7ab - 6b^2 = (a-2b)*(5a+3b).       <U>ANSWER</U>
</pre>


Solved.



<U>The lesson to learn from the solution is THIS</U> :


<pre>
    If you have difficulties factoring such a quadratic polynomial of 2 variables 
        (in other words, if you do not see the factoring formula immediately before your eyes in your mind)
     do the following (as I did in my solution)


        - reduce your quadratic polynomial of two variable to the quadratic polynomial of one new variable;

        - then find the roots of the reduced quadratic polynomial using the quadratic formula;

        - then factor the reduced quadratic polynomial using its roots;

        - then return back to the original variables "a" and "b".
</pre>


This method works ALWAYS like an army tank.



It means that &nbsp;<U>if the factoring formula does exist over rational (or integer) numbers</U>, 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>then the method provides / (guarantees) you will find the formula</U>.