What are the solutions to the equation x^8=-x^7+3 ?

The solutions to the equation x^8=-x^7+3 are x\approx 1.05549…,x\approx-1.35595…

Find the zeros of x^8=-x^7+3

The zeros of x^8=-x^7+3 are x\approx 1.05549…,x\approx-1.35595…

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x^{8} = - x^{7} + 3

x \approx 1.05549 \dots, x \approx - 1.35595 \dots

Solution steps

x^{8} = - x^{7} + 3

Move

3

to the left side

x^{8} - 3 = - x^{7}

Move

x^{7}

to the left side

x^{8} - 3 + x^{7} = 0

Write in the standard form

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

x^{8} + x^{7} - 3 = 0

Find one solution for

x^{8} + x^{7} - 3 = 0

using Newton-Raphson:

x \approx 1.05549 \dots

Apply long division:

\frac{x ^{8} + x ^{7} - 3}{x - 1.05549 \dots} = x^{7} + 2.05549 \dots x^{6} + 2.16957 \dots x^{5} + 2.28998 \dots x^{4} + 2.41707 \dots x^{3} + 2.55122 \dots x^{2} + 2.69281 \dots x + 2.84225 \dots

x^{7} + 2.05549 \dots x^{6} + 2.16957 \dots x^{5} + 2.28998 \dots x^{4} + 2.41707 \dots x^{3} + 2.55122 \dots x^{2} + 2.69281 \dots x + 2.84225 \dots \approx 0

Find one solution for

x^{7} + 2.05549 \dots x^{6} + 2.16957 \dots x^{5} + 2.28998 \dots x^{4} + 2.41707 \dots x^{3} + 2.55122 \dots x^{2} + 2.69281 \dots x + 2.84225 \dots = 0

using Newton-Raphson:

x \approx - 1.35595 \dots

Apply long division:

\frac{x ^{7} + 2.05549 \dots x ^{6} + 2.16957 \dots x ^{5} + 2.28998 \dots x ^{4} + 2.41707 \dots x ^{3} + 2.55122 \dots x ^{2} + 2.69281 \dots x + 2.84225 \dots}{x + 1.35595 \dots} = x^{6} + 0.69954 \dots x^{5} + 1.22102 \dots x^{4} + 0.63432 \dots x^{3} + 1.55696 \dots x^{2} + 0.44004 \dots x + 2.09612 \dots

x^{6} + 0.69954 \dots x^{5} + 1.22102 \dots x^{4} + 0.63432 \dots x^{3} + 1.55696 \dots x^{2} + 0.44004 \dots x + 2.09612 \dots \approx 0

Find one solution for

x^{6} + 0.69954 \dots x^{5} + 1.22102 \dots x^{4} + 0.63432 \dots x^{3} + 1.55696 \dots x^{2} + 0.44004 \dots x + 2.09612 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 1.05549 \dots, x \approx - 1.35595 \dots

3 x^{4} - 8 x^{3} + 2 = 0

- 2 x^{4} + 2 = 0

12 x - 4 x^{3} = 0

x^{4} - 10 x^{2} - 20 x - 16 = 0

x^{4} + 2 x^{3} - 10 x^{2} - 18 x + 9 = 0

What are the solutions to the equation x^8=-x^7+3 ?
The solutions to the equation x^8=-x^7+3 are x\approx 1.05549…,x\approx-1.35595…
Find the zeros of x^8=-x^7+3
The zeros of x^8=-x^7+3 are x\approx 1.05549…,x\approx-1.35595…