+0

# Solve 3x^3 - 21x^2 - 50x + 220 = 0

0
263
2

Solve 3x^3 - 21x^2 - 50x + 220 = 0

Guest Sep 23, 2014

### Best Answer

#2
+18827
+10

Solve 3x^3 - 21x^2 - 50x + 220 = 0

$$\begin{array}{c|l} \quad & \quad 3x^3-21x^2-50x+220=0 \quad |\; :3 \\ \quad & \quad \\ x^3+ax^2+bx+c=0 \quad & \quad x^3-7x^2-\frac{50}{3}x+\frac{220}{3}=0 \\ \quad & \quad \\ \quad & \quad a=-7; \; b=-\frac{50}{3}; \;c=\frac{220}{3} \\ \quad & \quad \\ B=\frac{3b-a^2}{3} \quad C=\frac{2a^3-9ab+27c}{27} \quad & \quad B=\frac{3(-\frac{50}{3})-(-7)^2}{3}=-\frac{99}{3}=-33 \quad \\ \quad & \quad \\ \quad & \quad C=\frac{2(-7)^3-9(-7)(-\frac{50}{3})+27(\frac{220}{3})}{27} =-\frac{-686-1050+1980}{27}=\frac{244}{27} \quad \\ \quad & \quad \\ d=\sqrt{-\frac{4}{3}B} \quad & \quad d=\sqrt{-\frac{4}{3}(-33)}=\sqrt{4*11}=2\sqrt{11}=6.633249581\\ \quad & \quad \\ q=\dfrac{C}{d^3}} \quad & \quad q=\frac{244}{27}\frac{1}{8(\sqrt{11})^3}}=\frac{61}{54}\frac{1}{(\sqrt{11})^3}=0.030963286\\ \quad & \quad \\ 4q <1 \;! \small{\text{ 3 real solutions }}\quad & \quad 4q = 0.123853144 < 1\;!\\ \quad & \quad \\ \phi\ensurement{^{\circ}}= \sin^{-1}{(4q)} \quad & \quad \phi\ensurement{^{\circ}}= \sin^{-1}{(0.123853144)} \\ \quad & \quad \\ \quad & \quad \phi\ensurement{^{\circ}}=7.114531178\ensurement{^{\circ}}\\ \quad & \quad \\ x_1=d*sin{(\frac{1}{3}\phi\ensurement{^{\circ}})}-\frac{a}{3} \quad & \quad x_1=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}*7.11451178\ensurement{^{\circ}} ) }=\frac{7}{3}+2\sqrt{11}*0.041378847 \\ \quad & \quad x_1=2.607809554\\ \quad & \quad \\ x_2=d*sin{(\frac{1}{3} (\phi\ensurement{^{\circ}} +360\ensurement{^{\circ}} ) )}-\frac{a}{3} \quad & \quad x_2=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}* (7.11451178\ensurement{^{\circ}}+360\ensurement{^{\circ}}) ) }=\frac{7}{3}+2\sqrt{11}*0.844594254 \\ \quad & \quad x_2=7.935737816\\ \quad & \quad \\ x_3=d*sin{(\frac{1}{3} (\phi\ensurement{^{\circ}} +720\ensurement{^{\circ}} ) )}-\frac{a}{3} \quad & \quad x_3=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}* (7.11451178\ensurement{^{\circ}}+720\ensurement{^{\circ}}) ) }=\frac{7}{3}+2\sqrt{11}*(-0.885973101) \\ \quad & \quad x_3=-3.543547371\\ \end{array}$$

heureka  Sep 24, 2014
Sort:

### 2+0 Answers

#1
+80865
+5

3x^3 - 21x^2 - 50x + 220 = 0

The easiest way to solve this is by graphing.......see it here: https://www.desmos.com/calculator/nklgal58ln

Notice that we have 3 "real" roots.......the only other possibility would have been 1 real root and 2 non-real roots, since this is a 3rd degree polynomial.

CPhill  Sep 23, 2014
#2
+18827
+10
Best Answer

Solve 3x^3 - 21x^2 - 50x + 220 = 0

$$\begin{array}{c|l} \quad & \quad 3x^3-21x^2-50x+220=0 \quad |\; :3 \\ \quad & \quad \\ x^3+ax^2+bx+c=0 \quad & \quad x^3-7x^2-\frac{50}{3}x+\frac{220}{3}=0 \\ \quad & \quad \\ \quad & \quad a=-7; \; b=-\frac{50}{3}; \;c=\frac{220}{3} \\ \quad & \quad \\ B=\frac{3b-a^2}{3} \quad C=\frac{2a^3-9ab+27c}{27} \quad & \quad B=\frac{3(-\frac{50}{3})-(-7)^2}{3}=-\frac{99}{3}=-33 \quad \\ \quad & \quad \\ \quad & \quad C=\frac{2(-7)^3-9(-7)(-\frac{50}{3})+27(\frac{220}{3})}{27} =-\frac{-686-1050+1980}{27}=\frac{244}{27} \quad \\ \quad & \quad \\ d=\sqrt{-\frac{4}{3}B} \quad & \quad d=\sqrt{-\frac{4}{3}(-33)}=\sqrt{4*11}=2\sqrt{11}=6.633249581\\ \quad & \quad \\ q=\dfrac{C}{d^3}} \quad & \quad q=\frac{244}{27}\frac{1}{8(\sqrt{11})^3}}=\frac{61}{54}\frac{1}{(\sqrt{11})^3}=0.030963286\\ \quad & \quad \\ 4q <1 \;! \small{\text{ 3 real solutions }}\quad & \quad 4q = 0.123853144 < 1\;!\\ \quad & \quad \\ \phi\ensurement{^{\circ}}= \sin^{-1}{(4q)} \quad & \quad \phi\ensurement{^{\circ}}= \sin^{-1}{(0.123853144)} \\ \quad & \quad \\ \quad & \quad \phi\ensurement{^{\circ}}=7.114531178\ensurement{^{\circ}}\\ \quad & \quad \\ x_1=d*sin{(\frac{1}{3}\phi\ensurement{^{\circ}})}-\frac{a}{3} \quad & \quad x_1=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}*7.11451178\ensurement{^{\circ}} ) }=\frac{7}{3}+2\sqrt{11}*0.041378847 \\ \quad & \quad x_1=2.607809554\\ \quad & \quad \\ x_2=d*sin{(\frac{1}{3} (\phi\ensurement{^{\circ}} +360\ensurement{^{\circ}} ) )}-\frac{a}{3} \quad & \quad x_2=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}* (7.11451178\ensurement{^{\circ}}+360\ensurement{^{\circ}}) ) }=\frac{7}{3}+2\sqrt{11}*0.844594254 \\ \quad & \quad x_2=7.935737816\\ \quad & \quad \\ x_3=d*sin{(\frac{1}{3} (\phi\ensurement{^{\circ}} +720\ensurement{^{\circ}} ) )}-\frac{a}{3} \quad & \quad x_3=\frac{7}{3}+2\sqrt{11}*\sin{ (\frac{1}{3}* (7.11451178\ensurement{^{\circ}}+720\ensurement{^{\circ}}) ) }=\frac{7}{3}+2\sqrt{11}*(-0.885973101) \\ \quad & \quad x_3=-3.543547371\\ \end{array}$$

heureka  Sep 24, 2014

### 12 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