Question

Which equation is y=(x+3)^2+(x+4)^2 rewritten in vertex form?

Answer 1

`y-1/2=2(x+7/2)^2`

Explanation:

Equation of the Quadratic Function

The vertex form of the quadratic function has the following equation:

`y-k=a(x-h)^2`

Where (h, k) is the vertex of the parabola that results when plotting the function, and a is a coefficient different from zero.

The given function is:

`y=(x+3)^2+(x+4)^2`

Expanding the squares:

`y=x^2+6x+9+x^2+8x+16`

Simplifying:

`y=2x^2+14x+25`

Factoring 2 from the first two terms:

`y=2(x^2+7x)+25`

The expression in parentheses must be completed to form the square of a binomial:

`y=2(x^2+7x+49/4-49/4)+25`

Isolating the square of the binomial:

`y=2(x^2+7x+49/4)-49/2+25`

Operating and Factoring:

`y=2(x+7/2)^2+1/2`

Rearranging:

`\boxed{y-1/2=2(x+7/2)^2}`

Comparing with the vertex form:

a=2, h=-7/2, k=1/2