Rewrite (a quartic) equation as a square

Question

Given the equation

$$\frac{u^4}{3} - 2u^3 + \frac{23}{3}u^2 - 6u + 8$$

I want to rewrite it in the form $x^2 + 7$.

As it is a quartic, I started by letting $x = au^2 + bu + c$, since squaring this $x$ will give the desired powers for $u$.

Subbing in for $x$

$$(au^2 + bu + c)^2 + 7$$

$$= a^2u^4 + 2abu^3 + (2ac + b^2)u^2 + 2bcu + c^2 + 7$$

Then I let the coefficients in both equations be equal yielding $a^2 = \frac{1}{3}$, $2ab = -2$, $2ac + b^2 = \frac{23}{3}$, and $c^2 + 7 = 8$.

The issue lies here, the system of equations seems to be inconsistent.

Solving for the first 3 gives the values $a = \pm \frac{1}{\sqrt{3}}$, $b = \mp \sqrt{3}$, and $c = \pm \frac{7}{\sqrt{3}}$. However, this value of c disagrees with the last equation above $c^2 + 7 = 8 \implies c = \pm 1$, and likewise for the second last equation.

Is there any way to rewrite the above equation as a square, or is this enough to show it is not possible?

EDIT As a follow up question, how would I rewrite it in the general form $x^2 + D$? What values of D are possible?

If your calculations are right then you cannot write in the given form. But you could write the given expression in the form of $(ax^2+bx+c)^2+d$. — Math Lover, Jan 23 '18 at 18:34
Why do you want to write it as $x^2+7$ as first place? What is the importance of $7$? — Weijun Zhou, Jan 23 '18 at 18:34
@WeijunZhou I suppose there isn't too much significance attached to 7, any number there would do. — spyr03, Jan 23 '18 at 18:47

Andy Walls · Accepted Answer · 2018-01-28T15:47:34.167

It can't generally be written as $x^2 +D$, as the inconsistent solution you found in your approach demonstrates.

Your approach develops an over determined system of equations, resulting in 5 equations, with only 4 unknowns (a,b,c, and d). It may work in special cases, but you need another degree of freedom to make it work in the general case.

You should be able to make the following always work $$\left(au^2+bu+c\right)^2+ du + e$$

For your quartic, a rewritten expression of that form is $$\left(\dfrac{u^2}{\sqrt{3}} - \sqrt{3}u+ \dfrac{7}{\sqrt{3}}\right)^2 +8u -\dfrac{25}{3}$$

FWIW, the form $x^2+D$ imposes a special sort of symmetry on the quartic, i.e. the first and third turning points of the quartic have the same value on the vertical axis. A general quartic does not have that symmetry.

Rewrite (a quartic) equation as a square

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