Alright, let's go through each given expression and the process to determine the number that must be added to make each expression a perfect square trinomial. I'll then express each perfect square trinomial as a square of a binomial.
### 1. [tex]\( x^2 + 2x + \)[/tex]
To make this a perfect square trinomial, we need to complete the square:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\( 2 \)[/tex].
- Divide it by [tex]\( 2 \)[/tex] to get [tex]\( \frac{2}{2} = 1 \)[/tex].
- Square the result to get [tex]\( 1^2 = 1 \)[/tex].
So, the number that must be added is [tex]\( 1 \)[/tex].
Thus, [tex]\( x^2 + 2x + 1 = (x + 1)^2 \)[/tex].
### 2. [tex]\( m^2 + 6m + \)[/tex]
To make this a perfect square trinomial, we complete the square:
- Take the coefficient of [tex]\( m \)[/tex], which is [tex]\( 6 \)[/tex].
- Divide it by [tex]\( 2 \)[/tex] to get [tex]\( \frac{6}{2} = 3 \)[/tex].
- Square the result to get [tex]\( 3^2 = 9 \)[/tex].
So, the number that must be added is [tex]\( 9 \)[/tex].
Thus, [tex]\( m^2 + 6m + 9 = (m + 3)^2 \)[/tex].
### 3. [tex]\( a^2 - 4a + \)[/tex]
To make this a perfect square trinomial, we complete the square:
- Take the coefficient of [tex]\( a \)[/tex], which is [tex]\( -4 \)[/tex].
- Divide it by [tex]\( 2 \)[/tex] to get [tex]\( \frac{-4}{2} = -2 \)[/tex].
- Square the result to get [tex]\( (-2)^2 = 4 \)[/tex].
So, the number that must be added is [tex]\( 4 \)[/tex].
Thus, [tex]\( a^2 - 4a + 4 = (a - 2)^2 \)[/tex].
### 4. [tex]\( n^2 - 16n + \)[/tex]
To make this a perfect square trinomial, we complete the square:
- Take the coefficient of [tex]\( n \)[/tex], which is [tex]\( -16 \)[/tex].
- Divide it by [tex]\( 2 \)[/tex] to get [tex]\( \frac{-16}{2} = -8 \)[/tex].
- Square the result to get [tex]\( (-8)^2 = 64 \)[/tex].
So, the number that must be added is [tex]\( 64 \)[/tex].
Thus, [tex]\( n^2 - 16n + 64 = (n - 8)^2 \)[/tex].
### 5. [tex]\( y^2 + 10y + \)[/tex]
To make this a perfect square trinomial, we complete the square:
- Take the coefficient of [tex]\( y \)[/tex], which is [tex]\( 10 \)[/tex].
- Divide it by [tex]\( 2 \)[/tex] to get [tex]\( \frac{10}{2} = 5 \)[/tex].
- Square the result to get [tex]\( 5^2 = 25 \)[/tex].
So, the number that must be added is [tex]\( 25 \)[/tex].
Thus, [tex]\( y^2 + 10y + 25 = (y + 5)^2 \)[/tex].
In summary:
1. [tex]\( x^2 + 2x + 1 = (x + 1)^2 \)[/tex]
2. [tex]\( m^2 + 6m + 9 = (m + 3)^2 \)[/tex]
3. [tex]\( a^2 - 4a + 4 = (a - 2)^2 \)[/tex]
4. [tex]\( n^2 - 16n + 64 = (n - 8)^2 \)[/tex]
5. [tex]\( y^2 + 10y + 25 = (y + 5)^2 \)[/tex]
