+0

# a^2 + b^2 = 20; ab + 6b = 32; what are all the pairs (a; b)

0
367
3

a^2 + b^2 = 20 ab + 6b = 32 what are all the pairs (a; b)

Guest Apr 22, 2015

#2
+81154
+10

ab + 6b = 32    →  b(a + 6)  = 32  →   b = 32/(a + 6)   (2)

Substitue (2) into (1) and we  have

a^2 + [32/(a + 6)]^2  = 20      simplify

a^2(a + 6)^2 + 1024 = 20(a+6)^2

a^2(a^2 + 12a + 36) + 1024 = 20a^2 + 240a + 720

a^4 + 12a^3 + 36a^2 + 1024 = 20a^2 + 240a + 720

a^4 + 12a^3 + 16a^2 - 240a + 304  = 0

Using the on site solver we have one real soution for a

$${{\mathtt{a}}}^{{\mathtt{4}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{16}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{240}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\small\textbf+\,}}{\mathtt{304}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{8}}\\ {\mathtt{a}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{8}}\\ {\mathtt{a}} = {\mathtt{2}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{3.464\: \!101\: \!615\: \!137\: \!754\: \!6}}{i}\\ {\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3.464\: \!101\: \!615\: \!137\: \!754\: \!6}}{i}\\ {\mathtt{a}} = {\mathtt{2}}\\ \end{array} \right\}$$

And b =

32 /(a + 6)   = 32 /(2 + 6)  =  32 /8   = 4

CPhill  Apr 22, 2015
Sort:

#1
+26412
+10

You can do this on the calculator here using the "Equation" mode:

.

Alan  Apr 22, 2015
#2
+81154
+10

ab + 6b = 32    →  b(a + 6)  = 32  →   b = 32/(a + 6)   (2)

Substitue (2) into (1) and we  have

a^2 + [32/(a + 6)]^2  = 20      simplify

a^2(a + 6)^2 + 1024 = 20(a+6)^2

a^2(a^2 + 12a + 36) + 1024 = 20a^2 + 240a + 720

a^4 + 12a^3 + 36a^2 + 1024 = 20a^2 + 240a + 720

a^4 + 12a^3 + 16a^2 - 240a + 304  = 0

Using the on site solver we have one real soution for a

$${{\mathtt{a}}}^{{\mathtt{4}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{16}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{240}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\small\textbf+\,}}{\mathtt{304}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{8}}\\ {\mathtt{a}} = {\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{8}}\\ {\mathtt{a}} = {\mathtt{2}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{3.464\: \!101\: \!615\: \!137\: \!754\: \!6}}{i}\\ {\mathtt{a}} = {\mathtt{\,-\,}}{\mathtt{8}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3.464\: \!101\: \!615\: \!137\: \!754\: \!6}}{i}\\ {\mathtt{a}} = {\mathtt{2}}\\ \end{array} \right\}$$

And b =

32 /(a + 6)   = 32 /(2 + 6)  =  32 /8   = 4

CPhill  Apr 22, 2015
#3
+91517
0

Thanks Alan and CPhill

A very nice display from each of you.:)

Melody  Apr 23, 2015

### 5 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details