en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

roots x^4-2x^3-5x^2+10x-1

Solution

roots

x^{4} - 2 x^{3} - 5 x^{2} + 10 x - 1

Solution

x \approx 0.10582 \dots, x \approx 1.71635 \dots, x \approx - 2.25918 \dots, x \approx 2.43700 \dots

Solution steps

x^{4} - 2 x^{3} - 5 x^{2} + 10 x - 1

The roots are the intercepts with the x-axis

(y = 0)

x^{4} - 2 x^{3} - 5 x^{2} + 10 x - 1 = 0

Find one solution for

x^{4} - 2 x^{3} - 5 x^{2} + 10 x - 1 = 0

using Newton-Raphson:

x \approx 0.10582 \dots

Apply long division:

\frac{x ^{4} - 2 x ^{3} - 5 x ^{2} + 10 x - 1}{x - 0.10582 \dots} = x^{3} - 1.89417 \dots x^{2} - 5.20044 \dots x + 9.44966 \dots

x^{3} - 1.89417 \dots x^{2} - 5.20044 \dots x + 9.44966 \dots \approx 0

Find one solution for

x^{3} - 1.89417 \dots x^{2} - 5.20044 \dots x + 9.44966 \dots = 0

using Newton-Raphson:

x \approx 1.71635 \dots

Apply long division:

\frac{x ^{3} - 1.89417 \dots x ^{2} - 5.20044 \dots x + 9.44966 \dots}{x - 1.71635 \dots} = x^{2} - 0.17781 \dots x - 5.50564 \dots

x^{2} - 0.17781 \dots x - 5.50564 \dots \approx 0

Find one solution for

x^{2} - 0.17781 \dots x - 5.50564 \dots = 0

using Newton-Raphson:

x \approx - 2.25918 \dots

Apply long division:

\frac{x ^{2} - 0.17781 \dots x - 5.50564 \dots}{x + 2.25918 \dots} = x - 2.43700 \dots

x - 2.43700 \dots \approx 0

x \approx 2.43700 \dots

The solutions are

x \approx 0.10582 \dots, x \approx 1.71635 \dots, x \approx - 2.25918 \dots, x \approx 2.43700 \dots

Popular Examples

x^{3} + x^{2} + 9 x + 9 = 0

z^{4} = - 1 + i

(x^2-10)^2-3(x^2-10)-4=0

(x^{2} - 10)^{2} - 3 (x^{2} - 10) - 4 = 0

2 x^{5} + 24 x = 14 x^{3}

1/2 =4x^3-9x^2-1

\frac{1}{2} = 4 x^{3} - 9 x^{2} - 1