Questions: Consider the quadratic function (y=3 x^2+30 x+65) Rewrite the function in vertex format.

Solution Steps

To rewrite the quadratic function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. We will manipulate the given equation to match this form.

The given quadratic function is in the standard form:

\[ y = ax^2 + bx + c \]

where \( a = 3 \), \( b = 30 \), and \( c = 65 \).

To rewrite the quadratic function in vertex form, we need to complete the square. The vertex form of a quadratic function is:

\[ y = a(x - h)^2 + k \]

First, factor out the coefficient of \( x^2 \) from the first two terms:

\[ y = 3(x^2 + 10x) + 65 \]

Next, complete the square inside the parentheses. To do this, take half of the coefficient of \( x \), square it, and add and subtract it inside the parentheses:

1. Take half of 10: \( \frac{10}{2} = 5 \).
2. Square it: \( 5^2 = 25 \).

Add and subtract 25 inside the parentheses:

\[ y = 3(x^2 + 10x + 25 - 25) + 65 \]

\[ y = 3((x + 5)^2 - 25) + 65 \]

Distribute the 3 and simplify the expression:

\[ y = 3(x + 5)^2 - 75 + 65 \]

\[ y = 3(x + 5)^2 - 10 \]

Final Answer