y² - 14y + 49 = 25
You are able to recognize the right side, 25, as the square of an integer,
since 25 = 5², from your knowledge of basic math.
You should now be able to recognize the left side, the trinomial
y² - 14y + 49, as the square of a binomial, from your knowledge of algebra.
That's because it has four properties:
1. The first term is a square, (y² is the square of y)
2. The third term +49 is also a square, (+49 is the square of 7)
3. The middle term, ignoring the sign, is twice the product of the square roots
of the first and third terms (-14y, ignoring the sign, 14y, is twice the
product of the square root of y², which is y, and the square root of 49,
which is 7, since 2·y·7 = 14y.)
4. It factors as the square of the sum of the square roots, using the
sign of the middle term as the sign of the second term of the binomial.
That's a lot to swallow, but it is to your advantage to be able to recognize
that the left side factors as (y - 7)². So we have
y² - 14y + 49 = 25
(y - 7)² = 25
Now we use the principle of square roots:
y - 7 = ±5
y = 7 ± 5
Using the + sign, y = 7 + 5 = 12
Using the - sign, y = 7 - 5 = 2
Edwin
