Express 4x^2+32x+55 in the form (ax+b)^2+c, where a, b and c are constants and a is positive?

3 Answers
Feb 15, 2018

4x^2+32x+55, when expressed in the form
(ax+b)^2+c, we have
4x^2+32x+55=(2x+8)^2-9

Explanation:

4x^2+32x+55
Here,
Changing
4x^2 to (2x)^2
32x to2(2x)(8)
we have
4x^2+32x+55=(2x)^2+2(2x)(8)+55
We have,
(2x)^2+2(2x)(8)+8^2=(2x+8)^2

Since 8^2=64

(2x)^2+2(2x)(8)+64=(2x+8)^2
and
changing now,
55=64-9+We have
4x^2+32x+55=(2x)^2+2(2x)(8)+64-9
4x^2+32x+55=(2x+8)^2-9

Feb 15, 2018

Is the question correct? The normal format for this question type is completing the square ->a(x+b)^2+c
Answering for a(x+b)^2+c

y=4(x+4)^2-9

Explanation:

color(blue)("Preamble")

You can change any equation you wish into the format you wish as long as you add to it a correction that if applied with transform it back again.

A bit like

2=2larr" Original structure"

3=2 larr" Changed and "color(red)("not true")

3-1=2 larr" Now we have 'forced' it to be true"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Answering the question")

Given ubrace(y=4x^2+32x+55 larr)" Starting point""
color(white)("ddddddddddddd")darr

ubrace("You should go directly from that to this:")
color(white)("dddddddddddddd")darr

color(white)("ddddd")y=4(x+4)^2+k+55

then determine the value of k

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(brown)("This is how I got there taking it one step at a time.")

I was challenged once on a point so I am doing it this way:

For each change we must include a correction. The value of this correction will be different as we proceed step-wise through the modifications on the path to the final format.

color(blue)(Stepcolor(white)("d") 1 => k_1)

y=(4x^2+32x)+k_1+55 where in the end k+55->c
At this point k_1=0 as we have not changed any values.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)(Step 2 => k_1) No values changed yet

Factor out the 4

y=4(x^2+32/4x)+k_1+55
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)(Stepcolor(white)("d") 3 => k_2) Values starting to change: Halve the 32/4x

y=4(x^2+32/8x)+k_2+55
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)(Stepcolor(white)("d")4 => k_3) Remove the x from 32/8x

y=4(x^2+32/8)+k_3+55
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)(Stepcolor(white)("d")5 => k_4) Move the 2 from x^2 to outside the brackets

y=4(x+32/8)^2+k_4+55

y=4(x+4)^2+k_4+55 larr" Nearly there!"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)(Stepcolor(white)("d")6 => k_4) Determine the value of k_4

y=color(green)(4)(x+color(magenta)(4))^2+k_4+55

Set (color(green)(4)xxcolor(magenta)(4^2) )+k_4=0 larr" The correction"

64+k_4=0=>k_4=-64 So substituting for k_4 we have:

y=4(x+4)^2+k_4+55 color(white)("ddd")->color(white)("dddd") y=4(x+4)^2-64+55

color(white)("dddddddddddddddddddddd")-> color(white)("dddd") y=4(x+4)^2-9

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Tony BTony B

Feb 15, 2018

Assuming your question is correct.

'Forcing' the completed square solution in my other answer into this format we have:

y=(sqrt(4)color(white)("d")x+4sqrt(4)color(white)(.))^2-9" " which gives:

y=(2x+8)^2-9

Explanation:

Expanding : y=(2x+8)^2-9

y=4x^2+32x+64-9

y=4x^2+32x+55

Tony BTony B