Complete the square to re-write the quadratic function in vertex form:

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

GPT-4o mini · Answer

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) $.