Video Transcript
Factor 625𝑥 to the fourth power
plus 64𝑦 to the fourth power fully by completing the square.
In this question, we are asked to
factor the expression by completing the square. As such, we need to manipulate the
given binomial to include a perfect square trinomial in the form 𝑎 squared plus two
𝑎𝑏 plus 𝑏 squared, as this factors to 𝑎 plus 𝑏 all squared. We begin by letting the two terms
in our expression be 𝑎 squared and 𝑏 squared, respectively. This means that 𝑎 is equal to the
square root of 625𝑥 to the fourth power. And since the square root of 625 is
25, 𝑎 is equal to 25𝑥 squared. In the same way, 𝑏 is equal to the
square root of 64𝑦 to the fourth power. And this is equal to eight 𝑦
squared.
Our next step is to find an
expression for two 𝑎𝑏. This is equal to two multiplied by
25𝑥 squared multiplied by eight 𝑦 squared. This is equal to 400𝑥 squared 𝑦
squared. It is this term we need to add to
our expression to create a perfect square trinomial. Since we are adding 400𝑥 squared
𝑦 squared, we also need to subtract this from the initial expression in order for
the expression to remain the same. We are now in a position where the
first three terms form a perfect square trinomial. And this factors to 25𝑥 squared
plus eight 𝑦 squared all squared. And the entire expression can be
written as shown.
Next, we notice that 400𝑥 squared
𝑦 squared is a perfect square. It can be written as 20𝑥𝑦 all
squared. Our expression simplifies to 25𝑥
squared plus eight 𝑦 squared all squared minus 20𝑥𝑦 all squared. This is written in the form 𝑐
squared minus 𝑑 squared, which is known as the difference of squares. And we know that this can be
factored to 𝑐 plus 𝑑 multiplied by 𝑐 minus 𝑑. Our expression can therefore be
written as 25𝑥 squared plus eight 𝑦 squared plus 20𝑥𝑦 multiplied by 25𝑥 squared
plus eight 𝑦 squared minus 20𝑥𝑦. Rearranging the order of the terms,
we have that 625𝑥 to the fourth power plus 64𝑦 to the fourth power is equal to
25𝑥 squared plus 20𝑥𝑦 plus eight 𝑦 squared multiplied by 25𝑥 squared minus
20𝑥𝑦 plus eight 𝑦 squared. We could check this answer by
distributing the parentheses, where all terms apart from the first and last will
cancel.
