Which equation is $y=(x+3)^2+(x+4)^2$ rewritten in vertex form?
To rewrite the quadratic equation \(y = (x + 3)^2 + (x + 4)^2\) in vertex form, we first expand the squared terms. Applying the identity \((a+b)^2 = a^2 + 2ab + b^2\), we get:
\[(x + 3)^2 = x^2 + 6x + 9\]
\[(x + 4)^2 = x^2 + 8x + 16\]
Adding these together, the expression becomes:
\[y = (x^2 + 6x + 9) + (x^2 + 8x + 16) = 2x^2 + 14x + 25\]
Next, we convert the quadratic \(y = 2x^2 + 14x + 25\) into the vertex form \(y = a(x - h)^2 + k\) by completing the square. First, we factor out the leading coefficient \(2\) from the first two terms:
\[y = 2(x^2 + 7x) + 25\]
To complete the square inside the parentheses, we take half of the coefficient of \(x\) (which is \(7\)), square it (\((\frac{7}{2})^2 = \frac{49}{4} = 12.25\)), and add and subtract it:
\[y = 2(x^2 + 7x + 12.25 - 12.25) + 25\]
\[y = 2(x^2 + 7x + 12.25) - 2(12.25) + 25\]
\[y = 2(x + 3.5)^2 - 24.5 + 25\]
\[y = 2(x + 3.5)^2 + 0.5\]
The equation \(y = 2(x + 3.5)^2 + 0.5\) is the vertex form of the quadratic, where the vertex is at \((-3.5, 0.5)\). This concludes our step-by-step transformation. The final equation is \(\boxed{y = 2(x + 3.5)^2 + 0.5}\).
Vertex form is incredibly useful in real-life physics, particularly when analyzing the path of projectiles like a thrown ball. Since the vertex \((h, k)\) represents the highest or lowest point of the parabola, vertex form immediately tells us the maximum height of a ball or the minimum energy required for a mechanical system. By rewriting equations into this form, engineers and scientists can quickly identify peak performance parameters without needing to solve the entire quadratic equation.