To express each of the given numbers as a difference of squares, we need to find two integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[a^2 - b^2 = \text{given number}\][/tex]
Recall the identity:
[tex]\[a^2 - b^2 = (a + b)(a - b)\][/tex]
Let's solve for each number:
(a) 15
To write 15 as a difference of squares, we need to find [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[a^2 - b^2 = 15\][/tex]
After finding appropriate integers, we get:
[tex]\[4^2 - 1^2 = 16 - 1 = 15\][/tex]
Thus, the integers are [tex]\(a = 4\)[/tex] and [tex]\(b = 1\)[/tex]. So,
[tex]\[15 = 4^2 - 1^2\][/tex]
(b) 117
To write 117 as a difference of squares, we look for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[a^2 - b^2 = 117\][/tex]
With correct integers, we find:
[tex]\[59^2 - 58^2 = 3481 - 3364 = 117\][/tex]
Thus, the integers are [tex]\(a = 59\)[/tex] and [tex]\(b = 58\)[/tex]. So,
[tex]\[117 = 59^2 - 58^2\][/tex]
(c) 231
To write 231 as a difference of squares, we look for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[a^2 - b^2 = 231\][/tex]
With correct integers, we get:
[tex]\[16^2 - 5^2 = 256 - 25 = 231\][/tex]
Thus, the integers are [tex]\(a = 16\)[/tex] and [tex]\(b = 5\)[/tex]. So,
[tex]\[231 = 16^2 - 5^2\][/tex]
In summary:
- (a) 15 can be written as [tex]\(4^2 - 1^2\)[/tex].
- (b) 117 can be written as [tex]\(59^2 - 58^2\)[/tex].
- (c) 231 can be written as [tex]\(16^2 - 5^2\)[/tex].
