What are the solutions to the equation 1-4z+4z^4=0 ?

The solutions to the equation 1-4z+4z^4=0 are z\approx 0.25417…,z\approx 0.89679…

Find the zeros of 1-4z+4z^4=0

The zeros of 1-4z+4z^4=0 are z\approx 0.25417…,z\approx 0.89679…

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1 - 4 z + 4 z^{4} = 0

z \approx 0.25417 \dots, z \approx 0.89679 \dots

Solution steps

1 - 4 z + 4 z^{4} = 0

Write in the standard form

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

4 z^{4} - 4 z + 1 = 0

Find one solution for

4 z^{4} - 4 z + 1 = 0

using Newton-Raphson:

z \approx 0.25417 \dots

Apply long division:

\frac{4 z ^{4} - 4 z + 1}{z - 0.25417 \dots} = 4 z^{3} + 1.01669 \dots z^{2} + 0.25841 \dots z - 3.93431 \dots

4 z^{3} + 1.01669 \dots z^{2} + 0.25841 \dots z - 3.93431 \dots \approx 0

Find one solution for

4 z^{3} + 1.01669 \dots z^{2} + 0.25841 \dots z - 3.93431 \dots = 0

using Newton-Raphson:

z \approx 0.89679 \dots

Apply long division:

\frac{4 z ^{3} + 1.01669 \dots z ^{2} + 0.25841 \dots z - 3.93431 \dots}{z - 0.89679 \dots} = 4 z^{2} + 4.60385 \dots z + 4.38710 \dots

4 z^{2} + 4.60385 \dots z + 4.38710 \dots \approx 0

Find one solution for

4 z^{2} + 4.60385 \dots z + 4.38710 \dots = 0

using Newton-Raphson:

No Solution for

z \in R

The solutions are

z \approx 0.25417 \dots, z \approx 0.89679 \dots

5 x^{3} = 7 x^{2}

z, c = \frac{h z ^{5}}{t}

x \frac{x}{2} \frac{x}{3} 2 = 2 x

6 q^{3} + 6 = 2 q^{3} + 506

(x - 1)^{5} = x^{2} (x - 3) + x

What are the solutions to the equation 1-4z+4z^4=0 ?
The solutions to the equation 1-4z+4z^4=0 are z\approx 0.25417…,z\approx 0.89679…
Find the zeros of 1-4z+4z^4=0
The zeros of 1-4z+4z^4=0 are z\approx 0.25417…,z\approx 0.89679…