Roots of a Quartic (Vieta's Formulas)

Question

Question: The quartic polynomial $x^4 −8x^3 + 19x^2 +kx+ 2$ has four distinct real roots denoted $a, b, c,d$ in order from smallest to largest. If $a + d = b + c$ then

(a) Show that $a + d = b + c = 4$.

(b) Show that $abcd = 2$ and $ad + bc = 3$.

(c) Find $ad$ and $bc.$

(d) Find $a, b, c, d$ and $k$.

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My attempt:

$$ x^4 −8x^3 + 19x^2 +kx+ 2 $$

With Vieta's formulas;

$$ a+b+c+d = 8 $$

$$ab+ac+ad+bc+bd+cd = 19$$

$$abc + abd + acd + bcd = -k$$

$$ abcd = 2 $$

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(a) Show that $a + d = b + c = 4$

As $$ a+b+c+d = 8 $$

but $b+c = a+d $

$$ 2a+2d = 8 $$

$$ a+d = 4 $$

Hence $$ a+d = b+c = 4 $$

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(b) Show that $abcd = 2$ and $ad + bc = 3$

As $$ abcd = 2 $$

and

$$ab+ac+ad+bc+bd+cd = 19$$

$$ ad+bc + a(b+c) + d(b+c) = 19 $$

$$ ad+bc + (a+d)(b+c) = 19 $$

$$ ad+bc + (4)(4) = 19 $$

$$ ad+bc + 16 = 19 $$

$$ ad +bc = 3 $$

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(c) Find $ad$ and $bc.$

Given $ad +bc = 3$ and $ abcd = 2 $

Hence $$ bc = \frac {2}{ad}$$

$$ad +bc = 3$$

$$ ad + \frac {2}{ad} = 3 $$

Let $ad = z $

$$ z^2 - 3z + 2 = 0 $$

$$ ad = 2 $$

$$ bc = 1 $$

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(d) Find $a, b, c, d$ and $k$.

Given

$$abc + abd + acd + bcd = -k$$

$$ ad(b+c)+ bc(a+d) = - k $$

But $b+c=a+d=4$

$$ 4ad+ 4bc = - k $$

$$ 4(ad+bc) = -k $$

$$ 4(3) = -k $$

$$ k = -12 $$

Now this is the part which I am stuck on.. How do I find $a,b,c,d$?

Answer 1

Hint If you know that $ad = 2$ and $a + d = 4$, then $$(x - a)(x - d) = x^2 - (a + d) x + ad = x^2 - 4 x + 2 ,$$ so finding $a, d$ is just finding the roots of that quadratic. Of course, finding $b, c$ is analogous.

Answer 2

You already have lots of equations, some of them non-linear, from where hopefully the coefficients will be found:

$$\begin{align*}&a+d=4=b+c\\&ad=2\,,\,\,bc=1\end{align*}$$

so for example (as clearly all coefficients are non-zero):

$$c=\frac1b\implies 4=b+c=b+\frac1b\implies b^2-4b+1=0\implies b_{1,2}=2\pm\sqrt3\implies$$

$$c=2\mp\sqrt3$$

You can do something similar with $\;a,d\;$ and $\;ad=2\implies d=\frac2a\;$ and etc. Later, you can use Viete's formulas (3)-(4) to fix the signs.