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Popular >

4x^7+7x^4+5x^2-10=x^5

Solution

4 x^{7} + 7 x^{4} + 5 x^{2} - 10 = x^{5}

Solution

x \approx 0.90084 \dots, x \approx - 1.08371 \dots, x \approx - 1.18872 \dots

Solution steps

4 x^{7} + 7 x^{4} + 5 x^{2} - 10 = x^{5}

Move

x^{5}

to the left side

4 x^{7} + 7 x^{4} + 5 x^{2} - 10 - x^{5} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

4 x^{7} - x^{5} + 7 x^{4} + 5 x^{2} - 10 = 0

Find one solution for

4 x^{7} - x^{5} + 7 x^{4} + 5 x^{2} - 10 = 0

using Newton-Raphson:

x \approx 0.90084 \dots

Apply long division:

\frac{4 x ^{7} - x ^{5} + 7 x ^{4} + 5 x ^{2} - 10}{x - 0.90084 \dots} = 4 x^{6} + 3.60336 \dots x^{5} + 2.24606 \dots x^{4} + 9.02335 \dots x^{3} + 8.12861 \dots x^{2} + 12.32260 \dots x + 11.10072 \dots

4 x^{6} + 3.60336 \dots x^{5} + 2.24606 \dots x^{4} + 9.02335 \dots x^{3} + 8.12861 \dots x^{2} + 12.32260 \dots x + 11.10072 \dots \approx 0

Find one solution for

4 x^{6} + 3.60336 \dots x^{5} + 2.24606 \dots x^{4} + 9.02335 \dots x^{3} + 8.12861 \dots x^{2} + 12.32260 \dots x + 11.10072 \dots = 0

using Newton-Raphson:

x \approx - 1.08371 \dots

Apply long division:

\frac{4 x ^{6} + 3.60336 \dots x ^{5} + 2.24606 \dots x ^{4} + 9.02335 \dots x ^{3} + 8.12861 \dots x ^{2} + 12.32260 \dots x + 11.10072 \dots}{x + 1.08371 \dots} = 4 x^{5} - 0.73147 \dots x^{4} + 3.03878 \dots x^{3} + 5.73018 \dots x^{2} + 1.91874 \dots x + 10.24323 \dots

4 x^{5} - 0.73147 \dots x^{4} + 3.03878 \dots x^{3} + 5.73018 \dots x^{2} + 1.91874 \dots x + 10.24323 \dots \approx 0

Find one solution for

4 x^{5} - 0.73147 \dots x^{4} + 3.03878 \dots x^{3} + 5.73018 \dots x^{2} + 1.91874 \dots x + 10.24323 \dots = 0

using Newton-Raphson:

x \approx - 1.18872 \dots

Apply long division:

\frac{4 x ^{5} - 0.73147 \dots x ^{4} + 3.03878 \dots x ^{3} + 5.73018 \dots x ^{2} + 1.91874 \dots x + 10.24323 \dots}{x + 1.18872 \dots} = 4 x^{4} - 5.48639 \dots x^{3} + 9.56061 \dots x^{2} - 5.63478 \dots x + 8.61697 \dots

4 x^{4} - 5.48639 \dots x^{3} + 9.56061 \dots x^{2} - 5.63478 \dots x + 8.61697 \dots \approx 0

Find one solution for

4 x^{4} - 5.48639 \dots x^{3} + 9.56061 \dots x^{2} - 5.63478 \dots x + 8.61697 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 0.90084 \dots, x \approx - 1.08371 \dots, x \approx - 1.18872 \dots

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