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To determine the number that should be added to each expression in order to form a perfect square trinomial and then factor the resulting trinomial, we follow these steps:
1. Identify the coefficient [tex]\( b \)[/tex] of the [tex]\( x \)[/tex] term.
2. Calculate [tex]\( \left(\frac{b}{2}\right)^2 \)[/tex].
3. Add this value to the expression.
4. The resulting trinomial can be factored into a perfect square form.
Let's proceed with each of the given expressions:
### 1. [tex]\( a^2 + 6a + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = 6 \)[/tex].
2. Calculate [tex]\( \left(\frac{6}{2}\right)^2 = 3^2 = 9 \)[/tex].
3. Add this value: [tex]\( a^2 + 6a + 9 \)[/tex].
4. The factored form is [tex]\( (a + 3)^2 \)[/tex].
So,
[tex]\[ a^2 + 6a + 9 = (a + 3)^2. \][/tex]
### 2. [tex]\( b^2 - 8b + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -8 \)[/tex].
2. Calculate [tex]\( \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 \)[/tex].
3. Add this value: [tex]\( b^2 - 8b + 16 \)[/tex].
4. The factored form is [tex]\( (b - 4)^2 \)[/tex].
So,
[tex]\[ b^2 - 8b + 16 = (b - 4)^2. \][/tex]
### 3. [tex]\( d^2 - 22d + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -22 \)[/tex].
2. Calculate [tex]\( \left(\frac{-22}{2}\right)^2 = (-11)^2 = 121 \)[/tex].
3. Add this value: [tex]\( d^2 - 22d + 121 \)[/tex].
4. The factored form is [tex]\( (d - 11)^2 \)[/tex].
So,
[tex]\[ d^2 - 22d + 121 = (d - 11)^2. \][/tex]
### 4. [tex]\( f^2 + 20f + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = 20 \)[/tex].
2. Calculate [tex]\( \left(\frac{20}{2}\right)^2 = 10^2 = 100 \)[/tex].
3. Add this value: [tex]\( f^2 + 20f + 100 \)[/tex].
4. The factored form is [tex]\( (f + 10)^2 \)[/tex].
So,
[tex]\[ f^2 + 20f + 100 = (f + 10)^2. \][/tex]
### 5. [tex]\( y^2 - 2y + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -2 \)[/tex].
2. Calculate [tex]\( \left(\frac{-2}{2}\right)^2 = (-1)^2 = 1 \)[/tex].
3. Add this value: [tex]\( y^2 - 2y + 1 \)[/tex].
4. The factored form is [tex]\( (y - 1)^2 \)[/tex].
So,
[tex]\[ y^2 - 2y + 1 = (y - 1)^2. \][/tex]
### 6. [tex]\( c^2 - 30c + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -30 \)[/tex].
2. Calculate [tex]\( \left(\frac{-30}{2}\right)^2 = (-15)^2 = 225 \)[/tex].
3. Add this value: [tex]\( c^2 - 30c + 225 \)[/tex].
4. The factored form is [tex]\( (c - 15)^2 \)[/tex].
So,
[tex]\[ c^2 - 30c + 225 = (c - 15)^2. \][/tex]
In summary, the numbers to be added and the factored forms are:
1. [tex]\( a^2 + 6a + 9 = (a + 3)^2 \)[/tex]
2. [tex]\( b^2 - 8b + 16 = (b - 4)^2 \)[/tex]
3. [tex]\( d^2 - 22d + 121 = (d - 11)^2 \)[/tex]
4. [tex]\( f^2 + 20f + 100 = (f + 10)^2 \)[/tex]
5. [tex]\( y^2 - 2y + 1 = (y - 1)^2 \)[/tex]
6. [tex]\( c^2 - 30c + 225 = (c - 15)^2 \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex] of the [tex]\( x \)[/tex] term.
2. Calculate [tex]\( \left(\frac{b}{2}\right)^2 \)[/tex].
3. Add this value to the expression.
4. The resulting trinomial can be factored into a perfect square form.
Let's proceed with each of the given expressions:
### 1. [tex]\( a^2 + 6a + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = 6 \)[/tex].
2. Calculate [tex]\( \left(\frac{6}{2}\right)^2 = 3^2 = 9 \)[/tex].
3. Add this value: [tex]\( a^2 + 6a + 9 \)[/tex].
4. The factored form is [tex]\( (a + 3)^2 \)[/tex].
So,
[tex]\[ a^2 + 6a + 9 = (a + 3)^2. \][/tex]
### 2. [tex]\( b^2 - 8b + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -8 \)[/tex].
2. Calculate [tex]\( \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 \)[/tex].
3. Add this value: [tex]\( b^2 - 8b + 16 \)[/tex].
4. The factored form is [tex]\( (b - 4)^2 \)[/tex].
So,
[tex]\[ b^2 - 8b + 16 = (b - 4)^2. \][/tex]
### 3. [tex]\( d^2 - 22d + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -22 \)[/tex].
2. Calculate [tex]\( \left(\frac{-22}{2}\right)^2 = (-11)^2 = 121 \)[/tex].
3. Add this value: [tex]\( d^2 - 22d + 121 \)[/tex].
4. The factored form is [tex]\( (d - 11)^2 \)[/tex].
So,
[tex]\[ d^2 - 22d + 121 = (d - 11)^2. \][/tex]
### 4. [tex]\( f^2 + 20f + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = 20 \)[/tex].
2. Calculate [tex]\( \left(\frac{20}{2}\right)^2 = 10^2 = 100 \)[/tex].
3. Add this value: [tex]\( f^2 + 20f + 100 \)[/tex].
4. The factored form is [tex]\( (f + 10)^2 \)[/tex].
So,
[tex]\[ f^2 + 20f + 100 = (f + 10)^2. \][/tex]
### 5. [tex]\( y^2 - 2y + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -2 \)[/tex].
2. Calculate [tex]\( \left(\frac{-2}{2}\right)^2 = (-1)^2 = 1 \)[/tex].
3. Add this value: [tex]\( y^2 - 2y + 1 \)[/tex].
4. The factored form is [tex]\( (y - 1)^2 \)[/tex].
So,
[tex]\[ y^2 - 2y + 1 = (y - 1)^2. \][/tex]
### 6. [tex]\( c^2 - 30c + \ \)[/tex]
1. Identify the coefficient [tex]\( b \)[/tex]: [tex]\( b = -30 \)[/tex].
2. Calculate [tex]\( \left(\frac{-30}{2}\right)^2 = (-15)^2 = 225 \)[/tex].
3. Add this value: [tex]\( c^2 - 30c + 225 \)[/tex].
4. The factored form is [tex]\( (c - 15)^2 \)[/tex].
So,
[tex]\[ c^2 - 30c + 225 = (c - 15)^2. \][/tex]
In summary, the numbers to be added and the factored forms are:
1. [tex]\( a^2 + 6a + 9 = (a + 3)^2 \)[/tex]
2. [tex]\( b^2 - 8b + 16 = (b - 4)^2 \)[/tex]
3. [tex]\( d^2 - 22d + 121 = (d - 11)^2 \)[/tex]
4. [tex]\( f^2 + 20f + 100 = (f + 10)^2 \)[/tex]
5. [tex]\( y^2 - 2y + 1 = (y - 1)^2 \)[/tex]
6. [tex]\( c^2 - 30c + 225 = (c - 15)^2 \)[/tex]
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