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Popular >

x^4+6x^3-9x^2-54x+81=0

Solution

x^{4} + 6 x^{3} - 9 x^{2} - 54 x + 81 = 0

Solution

x \approx 1.85410 \dots, x \approx 1.85410 \dots, x \approx - 4.85410 \dots, x = - \frac{1597}{329}

+1

Decimal Notation

x = 1.85410 \dots, x = 1.85410 \dots, x = - 4.85410 \dots, x = - 4.85410 \dots

Solution steps

x^{4} + 6 x^{3} - 9 x^{2} - 54 x + 81 = 0

Find one solution for

x^{4} + 6 x^{3} - 9 x^{2} - 54 x + 81 = 0

using Newton-Raphson:

x \approx 1.85410 \dots

Apply long division:

\frac{x ^{4} + 6 x ^{3} - 9 x ^{2} - 54 x + 81}{x - 1.85410 \dots} = x^{3} + 7.85410 \dots x^{2} + 5.56231 \dots x - 43.68690 \dots

x^{3} + 7.85410 \dots x^{2} + 5.56231 \dots x - 43.68690 \dots \approx 0

Find one solution for

x^{3} + 7.85410 \dots x^{2} + 5.56231 \dots x - 43.68690 \dots = 0

using Newton-Raphson:

x \approx 1.85410 \dots

Apply long division:

\frac{x ^{3} + 7.85410 \dots x ^{2} + 5.56231 \dots x - 43.68690 \dots}{x - 1.85410 \dots} = x^{2} + 9.70820 \dots x + 23.56230 \dots

x^{2} + 9.70820 \dots x + 23.56230 \dots \approx 0

Find one solution for

x^{2} + 9.70820 \dots x + 23.56230 \dots = 0

using Newton-Raphson:

x \approx - 4.85410 \dots

Apply long division:

\frac{x ^{2} + 9.70820 \dots x + 23.56230 \dots}{x + 4.85410 \dots} = x + 4.85410 \dots

x + 4.85410 \dots \approx 0

x \approx - 4.85410 \dots

Approximate to algebraic form

x = - \frac{1597}{329}

The solutions are

x \approx 1.85410 \dots, x \approx 1.85410 \dots, x \approx - 4.85410 \dots, x = - \frac{1597}{329}

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