Sure, let's rewrite the given quadratic equation [tex]\( y = (x+3)^2 + (x+4)^2 \)[/tex] in the vertex form.
First, expand the given quadratic equation (although the expanded form has been provided):
[tex]\[ y = (x + 3)^2 + (x + 4)^2 \][/tex]
When expanded and simplified, we get:
[tex]\[ y = 2x^2 + 14x + 25 \][/tex]
For a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], we can complete the square to convert it to vertex form [tex]\( a(x-h)^2 + k \)[/tex].
Given [tex]\( y = 2x^2 + 14x + 25 \)[/tex]:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] (which is 2) from the first two terms:
[tex]\[ y = 2(x^2 + 7x) + 25 \][/tex]
2. Complete the square for the expression inside the parenthesis:
- Take the coefficient of [tex]\( x \)[/tex], which is 7, divide by 2, and square it: [tex]\( \left(\frac{7}{2}\right)^2 = \frac{49}{4} \)[/tex]
- Rewrite the expression:
[tex]\[ y = 2\left(x^2 + 7x + \frac{49}{4} - \frac{49}{4}\right) + 25 \][/tex]
[tex]\[ y = 2\left(\left(x + \frac{7}{2}\right)^2 - \frac{49}{4}\right) + 25 \][/tex]
- Distribute the 2:
[tex]\[ y = 2\left(x + \frac{7}{2}\right)^2 - 2 \cdot \frac{49}{4} + 25 \][/tex]
[tex]\[ y = 2\left(x + \frac{7}{2}\right)^2 - \frac{49}{2} + 25 \][/tex]
[tex]\[ y = 2\left(x + \frac{7}{2}\right)^2 - \frac{49}{2} + \frac{50}{2} \][/tex]
[tex]\[ y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2} \][/tex]
So, the equation [tex]\( y = 2x^2 + 14x + 25 \)[/tex] rewritten in vertex form is:
[tex]\[ y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2} \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ y = 2 \left( x + \frac{7}{2} \right)^2 + \frac{1}{2} \][/tex]
