Case VI.

114. When a trinomial is a perfect square.

lint Since (x + y)^2 = x^2 + 2xy + y^2,

the factors of $x^2+2xy+y^2$ are $x+y$ and $x+y$.

lint Since (x - y)^2 = x^2 - 2xy + y^2,

the factors of $x^2-2xy+y^2$ are $x-y$ and $x-y$.

Therefore, a trinomial is a perfect square, if its first and last terms are perfect squares and positive, and its middle term is twice the product of their square roots.

find the factors of a trinomial when it is a perfect square, therefore,

Extract the square roots of the first and last terms, and connect these square roots by the sign of the middle term.

Thus, if we wish to find the square root of $16a^2-24ab+9b^2,$ we take the square roots of $16a^2$ and $9b^2$, which are $4a$ and $3b$, respectively, and connect these square roots by the minus sign, the sign of the middle term. The square root is therefore $4a-3b.$

Again, if we wish to find the square root of $25x^2+40xy+16y^2,$ we take the square roots of $25x^2$ and $16y^2$ and connect these roots by the plus sign, the sign of the middle term. The square root is therefore $5x+4y.$

Exercise 37.

Resolve into factors:

1. $4x^2+4xy+y^2$.

2. $x^2+6xy+9y^2$.

3. $x^2+16x+64$.

4. $x^2+10ax+25a^2$.

5. $a^2-16a+64$.

6. $a^2-10ab+25b^2$.

7. $c^2-6cd+9d^2$.

8. $4x^2-4x+1$.