To write the given quadratic equation [tex]\( y = 2x^2 + 8x + 3 \)[/tex] in vertex form, we proceed as follows:
1. Start with the given equation:
[tex]\[ y = 2x^2 + 8x + 3 \][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ y = 2(x^2 + 4x) + 3 \][/tex]
3. Complete the square inside the parentheses:
- To complete the square, we need to add and subtract a term inside the parentheses that makes the quadratic expression a perfect square.
- The term we need to add and subtract is [tex]\( (4 / 2)^2 = 4 \)[/tex].
[tex]\[ y = 2(x^2 + 4x + 4 - 4) + 3 \][/tex]
[tex]\[ y = 2((x + 2)^2 - 4) + 3 \][/tex]
4. Distribute the 2 and simplify:
[tex]\[ y = 2(x + 2)^2 - 8 + 3 \][/tex]
[tex]\[ y = 2(x + 2)^2 - 5 \][/tex]
So, the given quadratic equation in vertex form is:
[tex]\[ y = 2(x + 2)^2 - 5 \][/tex]
Hence, the correct answer is:
[tex]\[ y = 2(x + 2)^2 - 5 \][/tex]
