To express the quadratic expression [tex]\( 2x^2 + 8x + 3 \)[/tex] in the form [tex]\( 2(x + p)^2 + q \)[/tex], you follow these steps:
1. Factor out the 2 from the terms involving [tex]\( x \)[/tex]:
[tex]\[
2x^2 + 8x + 3 = 2(x^2 + 4x) + 3
\][/tex]
2. Complete the square for the expression inside the parentheses:
- Start with [tex]\( x^2 + 4x \)[/tex].
- Take half the coefficient of [tex]\( x \)[/tex] which is [tex]\( 4 \)[/tex], so half of 4 is 2.
- Square this result: [tex]\( 2^2 = 4 \)[/tex].
3. Rewrite the expression by adding and subtracting this square term inside the parentheses:
[tex]\[
2(x^2 + 4x + 4 - 4) + 3 = 2[(x + 2)^2 - 4] + 3
\][/tex]
4. Distribute the 2 through the completed square term:
[tex]\[
2[(x + 2)^2 - 4] = 2(x + 2)^2 - 2 \cdot 4 = 2(x + 2)^2 - 8
\][/tex]
5. Combine the constants:
[tex]\[
2(x + 2)^2 - 8 + 3 = 2(x + 2)^2 - 5
\][/tex]
6. Identify the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] based on the transformed expression:
- The expression in the form [tex]\( 2(x + p)^2 + q \)[/tex] is [tex]\( 2(x + 2)^2 - 5 \)[/tex].
- Therefore, [tex]\( p = 2 \)[/tex] and [tex]\( q = -5 \)[/tex].
In conclusion, the expression [tex]\( 2x^2 + 8x + 3 \)[/tex] can be written as:
[tex]\[
2(x + 2)^2 - 5
\][/tex]
