Is there a method to factor equations with two variables raised to the second power?I found the equation 2b2−ab−a2=0 on a problem and couldn't find a way to factor it. Is there any method to factor these types of equations?

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Is there a method to factor equations with two variables raised to the second power?I found the equation 2b2−ab−a2=0 on a problem and couldn&#039;t find a way to factor it. Is there any method to factor these types of equations?

jchordig1d5 · Accepted Answer

An easy way to do it is by defining a variable u=ab and then dividing by b2 on both sides (assuming b≠0). We will then get a quadratic in u e.g.2b2−ab−a2=02−ab−a2b2=02−u−u2=0u2+u−2=0(u+2)(u−1)=0u=1,−2a=b,−2bWe also have the potentially singular solution that results when b=0. We can easily see that the only solution of that form is a,b=0, which is included in our general solution.Another method to solve these types of equations where all terms are order 2 is to consider one variable as a &quot;constant&quot; and solve using quadratic formula (or if you are bold, you can try for a factorization). For example, if we treat b as a constant in the example you provided,a2+ba−2b2=0a=−b±b2+8b22a=−b±3b2a=b,−2bThis will be pretty similar in terms of difficulty as the former method. This method also has applications when solving equations in two variables where the equation is, say, a quartic in one variable and a quadratic in the other. You can treat the quartic variable as a constant and solve using quadratic equation. I can&#039;t find any examples at the moment, but I&#039;ll add them later.

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