Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, 3, x, squared, minus, 48, x, plus, 195
y=3x
2
−48x+195

Answer 1

To rewrite the quadratic function \( y = 3x^2 - 48x + 195 \) in vertex form by completing the square, follow these steps:

1. Factor out the coefficient of \( x^2 \) from the \( x^2 \) and \( x \) terms:
\[
y = 3(x^2 - 16x) + 195
\]

2. Complete the square for the expression \( x^2 - 16x \):
- Take half of the coefficient of \( x \) (which is \(-16\)), square it, and add and subtract it inside the parentheses. Half of \(-16\) is \(-8\), and \((-8)^2 = 64\).
\[
y = 3(x^2 - 16x + 64 - 64) + 195
\]
This simplifies to:
\[
y = 3((x - 8)^2 - 64) + 195
\]

3. Distribute the \( 3 \) and simplify:
\[
y = 3(x - 8)^2 - 192 + 195
\]
\[
y = 3(x - 8)^2 + 3
\]

The quadratic function in vertex form is:
\[
y = 3(x - 8)^2 + 3
\]

The vertex of the parabola is at the point \( (8, 3) \).