Complete the square to re-write the quadratic function in vertex form:
y=2x ^2+16x+27
To complete the square, we need to find the constant term that will make the quadratic expression a perfect square trinomial. We can accomplish this by taking half of the coefficient of the x-term and squaring it.
In this case, the coefficient of the x-term is 16, so half of 16 is 8. Squaring 8 gives us 64.
Next, we add 64 to both sides of the equation:
y + 64 = 2x^2 + 16x + 27 + 64
Simplifying the right side of the equation:
y + 64 = 2x^2 + 16x + 91
Now, to rewrite the quadratic function in vertex form, we factor out the coefficient of x^2 from the first two terms:
y + 64 = 2(x^2 + 8x) + 91
To complete the square within the parentheses, we need to add the square of half the coefficient of x to both sides of the equation. Half of 8 is 4, and 4 squared is 16.
Adding 16 to both sides:
y + 64 + 16 = 2(x^2 + 8x + 16) + 91 + 16
Simplifying both sides of the equation:
y + 80 = 2(x + 4)^2 + 107
Finally, we can rewrite the quadratic function in vertex form by subtracting 80 from both sides:
y = 2(x + 4)^2 + 27