What is the vertex form of #y=12.25x^2 - 52.5x +110.25?

Question

1755 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-13T10:54:36

$color(blue)( y=49/4(x- 15/7)^2 +216/4)$

Given:

$color(green)(y=12.25x^2-52.5x+110.25)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~
Write as:

$color(blue)(" "y=49/4x^2 -105/2x+ 441/4)$

$color(brown)("Factor out "49/4)$

$color(blue)(" "y=49/4(x^2- 30/7x) +441/4)$

$color(brown)("Consider just the right hand side")$

$color(brown)(Apply "1/2xx-30/7x = -15/7x)$

$color(blue)(" "49/4(x^2- 15/7x) +441/4)$

$color(brown)("Remove the "x" from " -15/7x)$

$color(blue)(" "49/4(x^2- 15/7) +441/4)$

$color(brown)("Move the index of 2 from "x^2" to outside the bracket")$

$color(blue)(" "49/4(x- 15/7)^2 +441/4)$

$color(brown)("Now add the correction that compensates for the error we")$
$color(brown)("introduced by changing the brackets content.")$

$color(brown)("Let "k" be a constant")$

$color(blue)(" "y=49/4(x- 15/7)^2 +441/4+ k$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider the $-15/7$ from inside the brackets

Then $(-15/7)^2+k=0$

$=> k = -49/4(15/7)^2 =-56 1/4=-225/4$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$color(brown)(y=49/4(x- 15/7)^2 +441/4+ k$

Becomes
$color(blue)( y=49/4(x- 15/7)^2 +441/4-225/4)$

$color(blue)( y=49/4(x- 15/7)^2 +216/4)$

Tony B