Answer
Here are the trinomials expressed in the form \( a(x + p)^2 + q \):
1. \( 2x^{2} + 4x + 6 = 2(x + 1)^{2} + 4 \)
2. \( 2x^{2} - 6x + 8 = 2\left(x - \frac{3}{2}\right)^{2} + \frac{7}{2} \)
3. \( 3x^{2} - 18x + 7 = 3(x - 3)^{2} - 20 \)
4. \( -2x^{2} + 2x - 8 = -2\left(x - \frac{1}{2}\right)^{2} - \frac{15}{2} \)
5. \( -3x^{2} + 6x + 2 = -3(x - 1)^{2} + 5 \)
6. \( 5x^{2} + 10x - 3 = 5(x + 1)^{2} - 8 \)
Solution
Sure! Let's complete the square for each of the given trinomials and express them in the form \( a(x + p)^2 + q \).
### (1) \( 2x^{2} + 4x + 6 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
2(x^{2} + 2x) + 6
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} + 2x = (x + 1)^{2} - 1
\]
3. **Substitute back and simplify:**
\[
2\left[(x + 1)^{2} - 1\right] + 6 = 2(x + 1)^{2} - 2 + 6 = 2(x + 1)^{2} + 4
\]
**Final Form:**
\[
2(x + 1)^{2} + 4
\]
---
### (2) \( 2x^{2} - 6x + 8 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
2(x^{2} - 3x) + 8
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} - 3x = \left(x - \frac{3}{2}\right)^{2} - \frac{9}{4}
\]
3. **Substitute back and simplify:**
\[
2\left[\left(x - \frac{3}{2}\right)^{2} - \frac{9}{4}\right] + 8 = 2\left(x - \frac{3}{2}\right)^{2} - \frac{9}{2} + 8 = 2\left(x - \frac{3}{2}\right)^{2} + \frac{7}{2}
\]
**Final Form:**
\[
2\left(x - \frac{3}{2}\right)^{2} + \frac{7}{2}
\]
---
### (3) \( 3x^{2} - 18x + 7 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
3(x^{2} - 6x) + 7
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} - 6x = (x - 3)^{2} - 9
\]
3. **Substitute back and simplify:**
\[
3\left[(x - 3)^{2} - 9\right] + 7 = 3(x - 3)^{2} - 27 + 7 = 3(x - 3)^{2} - 20
\]
**Final Form:**
\[
3(x - 3)^{2} - 20
\]
---
### (4) \( -2x^{2} + 2x - 8 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
-2(x^{2} - x) - 8
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} - x = \left(x - \frac{1}{2}\right)^{2} - \frac{1}{4}
\]
3. **Substitute back and simplify:**
\[
-2\left[\left(x - \frac{1}{2}\right)^{2} - \frac{1}{4}\right] - 8 = -2\left(x - \frac{1}{2}\right)^{2} + \frac{1}{2} - 8 = -2\left(x - \frac{1}{2}\right)^{2} - \frac{15}{2}
\]
**Final Form:**
\[
-2\left(x - \frac{1}{2}\right)^{2} - \frac{15}{2}
\]
---
### (5) \( -3x^{2} + 6x + 2 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
-3(x^{2} - 2x) + 2
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} - 2x = (x - 1)^{2} - 1
\]
3. **Substitute back and simplify:**
\[
-3\left[(x - 1)^{2} - 1\right] + 2 = -3(x - 1)^{2} + 3 + 2 = -3(x - 1)^{2} + 5
\]
**Final Form:**
\[
-3(x - 1)^{2} + 5
\]
---
### (6) \( 5x^{2} + 10x - 3 \)
1. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
5(x^{2} + 2x) - 3
\]
2. **Complete the square inside the parentheses:**
\[
x^{2} + 2x = (x + 1)^{2} - 1
\]
3. **Substitute back and simplify:**
\[
5\left[(x + 1)^{2} - 1\right] - 3 = 5(x + 1)^{2} - 5 - 3 = 5(x + 1)^{2} - 8
\]
**Final Form:**
\[
5(x + 1)^{2} - 8
\]
---
### Summary of All Results
1. \( 2x^{2} + 4x + 6 = 2(x + 1)^{2} + 4 \)
2. \( 2x^{2} - 6x + 8 = 2\left(x - \frac{3}{2}\right)^{2} + \frac{7}{2} \)
3. \( 3x^{2} - 18x + 7 = 3(x - 3)^{2} - 20 \)
4. \( -2x^{2} + 2x - 8 = -2\left(x - \frac{1}{2}\right)^{2} - \frac{15}{2} \)
5. \( -3x^{2} + 6x + 2 = -3(x - 1)^{2} + 5 \)
6. \( 5x^{2} + 10x - 3 = 5(x + 1)^{2} - 8 \)
These are the trinomials expressed in the completed square form \( a(x + p)^2 + q \).
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