Question Video: Factorizing by Completing the Square | Nagwa Question Video: Factorizing by Completing the Square | Nagwa

Question Video: Factorizing by Completing the Square Mathematics • Second Year of Preparatory School

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Factorize fully 9𝑥⁴ − 21𝑥²𝑦² + 4𝑦⁴.

04:57

Video Transcript

Factorize fully nine 𝑥 to the fourth power minus 21𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power.

We notice that the polynomial contains two perfect square terms: nine 𝑥 to the fourth power and four 𝑦 to the fourth power. So we will attempt to factor this expression by completing the square. To use this method, we need to manipulate the expression to include a perfect square trinomial in the form 𝑎 squared plus or minus two 𝑎𝑏 plus 𝑏 squared, which can be factored as 𝑎 plus or minus 𝑏 squared. In these trinomials, 𝑎 and 𝑏 may be variables, constants, or products of variables and constants.

In this example, if we take 𝑎 squared to be nine 𝑥 to the fourth power and 𝑏 squared to be four 𝑦 to the fourth power, then our value of 𝑎 is the square root of 𝑎 squared, which is equal to three 𝑥 squared. And our value of 𝑏 is the square root of 𝑏 squared, which is equal to two 𝑦 squared. Then, our middle term is equal to two 𝑎𝑏, or in some cases negative two 𝑎𝑏. Two 𝑎𝑏 comes out to two times three 𝑥 squared times two 𝑦 squared, which is 12𝑥 squared 𝑦 squared.

In our next step, we will introduce the two 𝑎𝑏 term into the original expression, while also moving the negative 21𝑥 squared 𝑦 squared term to the end of the expression. For any term we introduce into the expression, we must add the same term with the opposite sign. This way, we are effectively adding zero, which does not change the polynomial. In this case, the zero gets added to the polynomial in the form of 12𝑥 squared 𝑦 squared minus 12𝑥 squared 𝑦 squared.

Our expression with these new terms is nine 𝑥 to the fourth power plus 12𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power minus 21𝑥 squared 𝑦 squared minus 12𝑥 squared 𝑦 squared.

We can now factor the first three terms as a perfect square trinomial, giving us three 𝑥 squared plus two 𝑦 squared squared. Then, we can combine the like terms, giving us negative 33𝑥 squared 𝑦 squared. Now at this step, we expect to have a difference of squares. However, 33 is not a perfect square. When this happens, we can try forming a perfect square trinomial with a negative two 𝑎𝑏 term instead of the positive two 𝑎𝑏 term we chose. So we will erase a bit of our work to return to the previous step.

We proceed to change the sign in front of the two 𝑎𝑏 term to a negative. This means our new expression is nine 𝑥 to the fourth power minus 12𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power minus 21𝑥 squared 𝑦 squared plus 12𝑥 squared 𝑦 squared. Then, we again factor the first three terms as a perfect square trinomial, but this time the factors are of the form 𝑎 minus 𝑏.

Now when we combine the like terms, we have negative nine 𝑥 squared 𝑦 squared. This is a difference of squares since the expression within the parentheses is being squared and nine 𝑥 squared 𝑦 squared is a perfect square, specifically the square of three 𝑥𝑦, where 𝑎 is in the first parentheses and 𝑏 is in the second parentheses.

Following the formula for factoring a difference of squares, we get three 𝑥 squared minus two 𝑦 squared minus three 𝑥𝑦 times three 𝑥 squared minus two 𝑦 squared plus three 𝑥𝑦.

Finally, we need to check whether the resulting polynomials within each set of parentheses can be factored. In this case, both polynomials are prime. So we have that three 𝑥 squared minus three 𝑥𝑦 minus two 𝑦 squared times three 𝑥 squared plus three 𝑥𝑦 minus two 𝑦 squared represents the full factorization of nine 𝑥 to the fourth power minus 21𝑥 squared 𝑦 squared plus four 𝑦 to the fourth power.

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