Question Video: Factoring a Quartic by Completing the Square Mathematics

Video Transcript

Factor four 𝑥 squared times 𝑥 squared minus six 𝑦 squared plus nine 𝑦 to the power of four by completing the square.

Let’s start by multiplying out the first term to see what we’re working with. Notice that we have a pair of perfect squares here. Recall the standard expansion of the binomial 𝑎 plus 𝑏 squared as 𝑎 squared plus two 𝑎𝑏 plus 𝑏 squared. This yields the equation 𝑎 squared plus 𝑏 squared equals 𝑎 plus 𝑏 squared minus two 𝑎𝑏.

We’re going to set 𝑎 squared to four 𝑥 to the four and 𝑏 squared to nine 𝑦 to the four. This gives 𝑎 equals two 𝑥 squared and 𝑏 equals three 𝑦 squared. Multiplying 𝑎 and 𝑏 together and then by two, we have two 𝑎𝑏 equals 12𝑥 squared 𝑦 squared. Using the starred identity, we have four 𝑥 to the power four plus nine 𝑦 to the power four equals two 𝑥 squared plus three 𝑦 squared all squared minus 12𝑥 squared 𝑦 squared.

We can now make this substitution in our expression. The sum of four 𝑥 to the fourth and nine 𝑦 to the fourth becomes two 𝑥 squared plus three 𝑦 squared all squared minus 12𝑥 squared 𝑦 squared. And the 24𝑥 squared 𝑦 squared term carries over. Collecting like terms, we have two 𝑥 squared plus three 𝑦 squared all squared minus 36𝑥 squared 𝑦 squared.

Notice that this second term, 36𝑥 squared 𝑦 squared, is itself a perfect square. We have transformed our original expression into a difference of two squares. We factor this in the normal way as the product of the sum and the difference. And this is as far as we can factor this expression over the rationals. As an aside, observe that if we allow irrational coefficients, we can actually break it into linear factors using quadratic formula.